Exponential models, transformations, and Log Mode

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Difficulty: Advanced
Estimated time: 75 minutes
Domains: Advanced Math Problem Solving and Data Analysis
Techniques: ExponentialsTransformationsGrowth-decayLog-modeRegressionEquation-solvingMethod-choice

What you’ll learn

Read shifted and scaled exponential models from the inside out.
Identify the reference input, baseline, interval multiplier, and per-unit multiplier.
Convert a growth or decay factor into a percent change.
Rewrite an exponential model in standard form and verify equivalent forms.
Solve exponential targets by graphing both sides.
Distinguish automatic Log Mode from ordinary nonlinear regression.

Prerequisites

Why this matters on the SAT

Read the multiplier the exponent actually describes

SAT exponential models often hide important information inside a shifted or scaled exponent. A base may describe a change every $5$ minutes rather than every minute, and a vertical shift may mean the factor applies to the amount above a baseline rather than to the entire output.

SAT example

After a ventilation system is turned on, the estimated concentration of a gas in a parking garage is modeled by

y=48(0.84)^{\frac{x-10}{5}}+7,

where $y$ is the concentration, in parts per million, and $x$ is the number of minutes after the system is turned on, with $x\ge10$ .

For each increase of $1$ minute in $x$ , the amount by which the concentration exceeds $7$ parts per million is multiplied by a constant factor. Which choice is closest to that factor?

A
$0.84$
B
$0.93$
C
$0.97$
D
$0.99$

Fast Desmos solution

The exponent increases by $1$ after $5$ minutes, so $0.84$ is the factor for a $5$ -minute interval. The per-minute factor is

0.84^{1/5}.

Enter 0.84^(1/5). Desmos returns approximately

0.9657,

which is closest to $0.97$ . The answer is C.

The factor applies to $y-7$ , the amount above the $7$ -parts-per-million baseline. It does not multiply the complete value of $y$ .

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The base 0.84 applies every 5 minutes, so the per-minute factor is the fifth root of 0.84.

Choose the job before choosing the tool

Use the information supplied by the question before choosing a calculator method.

Work from a given model when…

Evaluate directly when the equation is given and the question asks for an output, multiplier, or percent.
Graph both sides when the unknown is in the exponent and matching powers is not immediate.
Compare graphs, then prove exactly when the question asks which transformed expression is equivalent.

Fit parameters from data when…

Use exponential regression when a table supplies several input-output pairs and the prompt or pattern supports an exponential model.
Treat measured values as estimates. A fitted base comes from the complete data set, not one rounded ratio.

Use short algebra when the bases already match or a visible ratio answers the question faster.

Regression is for fitting parameters. A given function does not need to be fitted again.

Check your understanding:

One question gives $P(t)=300(1.04)^t$ and asks for $P(6)$ . Another supplies five measured pairs and states that an exponential model is appropriate. Which Desmos method fits each question?

1. Read every layer of the model

A useful transformed form is

f(x)=c+ab^{\frac{x-h}{k}},

where $k>0$ , $b>0$ , and $b\ne1$ .

Read it from the inside out:

$h$ is the reference input. At $x=h$ , the exponent is $0$ .
$c$ is the baseline added to the exponential part.
$a$ is the signed vertical offset from the baseline when $x=h$ , because $f(h)=c+a$ . The distance is $\lvert a\rvert$ .
$k$ is the input interval that increases the exponent by $1$ .
$b$ is the factor that multiplies $f(x)-c$ whenever $x$ increases by $k$ .

For the opening model,

y-7=48(0.84)^{\frac{x-10}{5}}.

At $x=10$ , the concentration is $7+48=55$ parts per million. Every additional $5$ minutes multiplies the amount above $7$ by $0.84$ .

When $a>0$ and $0<b<1$ , the exponential part decays and the graph approaches the horizontal asymptote $y=c$ . When $a>0$ and $b>1$ , $f(x)-c$ grows. If $a<0$ and $b>1$ , the magnitude $\lvert f(x)-c\rvert$ grows while the function moves downward.

Check your understanding:

The function $p(t)=50(0.6)^{t/5}$ models a mass in milligrams after $t$ days. What does the factor $0.6$ mean?

Common mistake:

Multiplying the entire output. In $c+ab^{(x-h)/k}$ , the base multiplies the exponential part $f(x)-c$ . It does not repeatedly multiply the fixed baseline $c$ .

2. Convert an interval factor into a per-unit factor

If $b$ is the multiplier every $k$ input units, then the per-unit multiplier is

r=b^{1/k}.

After $k$ one-unit steps,

r^k=\left(b^{1/k}\right)^k=b,

so the factors agree over the full interval.

For

p(t)=50(0.6)^{t/5},

the per-day multiplier is

0.6^{1/5}\approx0.9029.

This means the mass decreases by approximately

100(1-0.9029)\%=9.71\%

per day.

Use the multiplier to find the percent change:

\begin{aligned} \text{growth percent}&=100(r-1)\%, && r>1,\\ \text{decay percent}&=100(1-r)\%, && 0<r<1. \end{aligned}

Check your understanding:

A population is multiplied by $1.44$ every 2 hours. What is the hourly multiplier and the hourly percent increase?

Common mistake:

Dividing the percent change by the interval length. Exponential change compounds. Convert the percent to a multiplier, then take the appropriate root of that multiplier.

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The fifth root converts a five-day factor into a one-day factor; the complement gives the daily percent decrease.

3. Rewrite a transformed exponent into standard form

Equivalent exponential forms reveal different features. The form

ab^{\frac{x-h}{k}}+c

makes the reference input, interval length, and baseline visible. The form

A\cdot B^x+c

makes the per-unit multiplier $B$ visible.

In general,

\begin{aligned} c+ab^{\frac{x-h}{k}} &=c+ab^{-h/k}\left(b^{1/k}\right)^x\\ &=c+A\cdot B^x, \end{aligned}

where

A=ab^{-h/k}\qquad\text{and}\qquad B=b^{1/k}.

The vertical shift $c$ remains unchanged. The rewritten base $B$ is the per-unit multiplier.

Worked example

Consider

F(t)=72\cdot9^{\frac{t-2}{2}}.

Show that the function is equivalent to

G(t)=8\cdot3^t.

The graphs overlap, which is strong numerical evidence. Then confirm the equivalence exactly:

\begin{aligned} 72\cdot9^{\frac{t-2}{2}} &=72\cdot\left(3^2\right)^{\frac{t-2}{2}}\\ &=72\cdot3^{t-2}\\ &=72\cdot\frac{3^t}{9}\\ &=8\cdot3^t. \end{aligned}

The transformed form shows a factor of $9$ every $2$ input units. The standard form shows the per-unit multiplier $3$ .

Check your understanding:

Rewrite $40\cdot25^{(x-2)/2}$ in the form $A\cdot B^x$ .

Common mistake:

Moving part of the exponent outside the power. The subtraction and division in $b^{(x-h)/k}$ are inside the exponent. Rewrite the exponent as a sum or difference, then apply exponent rules to the complete power.

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The graphs overlap because both expressions simplify to 8 times 3 raised to t.

4. Solve an exponential target by graphing both sides

When the unknown appears in an exponent, graph the model and the target on separate lines. Their intersection gives the input that makes the outputs equal.

Worked example

Suppose

M(t)=120(0.85)^{t/4}

models a mass in grams after $t$ hours. After approximately how many hours will the mass reach $50$ grams?

Enter M(t)=120(0.85)^(t/4) and y=50. The graphs intersect near

(21.547,50).

Therefore, the model reaches $50$ grams after approximately

\boxed{21.5}

hours, rounded to the nearest tenth.

The graph point is numerical, so match the final rounding and units to the question. If a problem asks for the least whole-number time that meets a target, use the inequality direction and test the next allowable input instead of rounding automatically.

Try it yourself:

Change the target from y=50 to y=40. Predict whether the intersection will move left or right before selecting the new point, then reset the example.

Check your understanding:

The equation $200(0.5)^{t/6}=25$ has matching powers after division. Is graphing the shortest route, and what is $t$ ?

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The intersection shows when the exponential model first reaches the target value of 50 grams.

Do not confuse a logarithmic graph with Log Mode

Logarithmic equations are not part of the current Digital SAT Math content scope. This short calculator note prevents a vocabulary mix-up: a logarithm function and regression Log Mode are different ideas. A logarithmic equation can be solved by graphing each side and reading the intersection just as you would for another one-variable equation.

For example, to solve

\log_2(x+1)=3,

enter y=log_2(x+1) and y=3. The graphs intersect at

(7,3),

so $x=7$ . The equivalent exponential statement

2^3=x+1

confirms the result.

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The intersection solves the logarithmic equation without confusing it with regression Log Mode.

5. Use automatic Log Mode for exponential data

When a table contains positive outputs and the prompt supports a model of the form

y=ab^x,

enter the data in x_1 and y_1, then fit y_1~a*b^(x_1).

In the current College Board testing calculator, compatible exponential regressions use automatic Log Mode. Desmos applies a logarithmic transformation that turns the compatible model into a linear relationship before fitting $a$ and $b$ . You do not need to type logarithms or transform the table yourself.

For the exact table shown, Log Mode gives

a=90\qquad\text{and}\qquad b=0.8.

The fitted model is

y=90(0.8)^x.

The base $0.8$ means $80\%$ remains each input interval, so the output decreases by $20\%$ per interval.

Try it yourself:

Change the final output from $46.08$ to $45$ . Predict whether the fitted base will move above or below $0.8$ , inspect the result, then reset the table.

Automatic Log Mode does not decide whether an exponential model is appropriate. The prompt, context, and multiplicative pattern must support that choice.

It also has boundaries:

logarithmic transformation requires positive output values;
a zero or negative output prevents Log Mode, although Desmos may still return an ordinary nonlinear least-squares fit;
a model such as $ab^x+c$ with unknown $c$ cannot be linearized by the same transformation, so Desmos uses ordinary nonlinear regression instead; and
rounded measurements produce fitted estimates, not exact symbolic parameters.

The later lesson on regression failure modes examines residuals, exact data, and best-fit data in more depth.

Check your understanding:

A table contains an output value of $0$ , and a student wants to fit $y=ab^x$ with Log Mode. What should the student notice first?

Common mistake:

Using Log Mode to solve an equation. Log Mode fits compatible regression models to table data. It is not a separate command for solving an exponential equation. For a given equation with an unknown exponent, graph both sides or use matching powers.

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Automatic Log Mode fits the positive exponential data with an initial value of 90 and a multiplier of 0.8.

6. Example: Read the model, graph the target, and respect the domain

Worked example

The temperature $T(t)$ , in degrees Celsius, of a liquid $t$ minutes after cooling begins is modeled by

T(t)=18+72(0.75)^{t/6}.

Which choice is the

least integer value of $t$

for which $T(t)\le40$ ?

A
$24$
B
$25$
C
$30$
D
$31$

Step 1

Read the transformed model

The fixed baseline is $18$ . At $t=0$ , the temperature is

18+72=90

degrees Celsius. Every $6$ minutes, the amount above $18$ is multiplied by $0.75$ .

Step 2

Graph the model and target

Enter T(t)=18+72(0.75)^(t/6) and y=40. The graphs intersect at approximately

(24.728,40).

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The numerical intersection occurs after minute 24 but before minute 25.

Step 3

Apply the least-integer condition

The temperature is still above the target at $t=24$ :

T(24)=40.78125.

At $t=25$ ,

T(25)\approx39.715,

so the first integer time for which $T(t)\le40$ is

\boxed{25}.

The answer is B.

Common mistake:

Dropping the decimal part of the intersection. The intersection is after minute 24. Because the temperature is decreasing and must be at or below the target, test minute 25 rather than rounding down.

Finish the solution

The measured data are already entered. Add the exponential regression, use the fitted base, and submit the percent decrease per interval.

Turn the fitted base into a percent

Finish the solution

A scientist measures a sample once every $4$ hours. Let $n$ be the number of $4$ -hour intervals after the first measurement.

$n$	$0$	$1$	$2$	$3$
Mass (milligrams)	$900$	$744$	$620$	$511$

An exponential model of the form

M(n)=ab^n

is used to fit the measured data. According to the fitted model, by what percent does the sample's mass decrease every $4$ hours, rounded to the nearest tenth of a percent?

First steps

The table is already loaded with interval number in x_1 and mass in y_1.
Enter y_1~a*b^(x_1).
Use the fitted value of $b$ without retyping its displayed decimal.

Finish it

Your answer

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Add the exponential regression, then convert its fitted base into the percent decrease per 4-hour interval.

Practice problems

SAT practice problems

Identify what the exponent and base mean before calculating. Use direct evaluation, graph overlap, graphing both sides, or regression according to the information supplied.

Interpret an interval factor

Practice problem

The concentration $m(t)$ , in parts per million, of a substance $t$ days after a treatment begins is modeled by

m(t)=12+80(0.72)^{t/6}.

Which choice best describes the meaning of the factor $0.72$ ?

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Evaluate the model at inputs 6 units apart and compare each output with the baseline 12.

Rewrite a transformed exponential

Practice problem

A company models the amount of data, in gigabytes, stored in an archive after $r$ weeks with

D(r)=96\cdot4^{\frac{3r-2}{2}}.

Which choice gives an equivalent function in the form $D(r)=A\cdot B^r$ , where $A$ and $B$ are constants?

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Graph the original expression and the choices to identify overlap, then confirm the selected form with exponent rules.

Find the least allowable input

Practice problem

The function

S(t)=600(1.08)^{t/3}

models the number of subscribers on a channel $t$ days after a promotion begins. Which choice is the least integer value of $t$ for which

S(t)\ge900?

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Graph the model and the target, then apply the least-integer condition to the numerical intersection.

Practice 4 uses the multi-condition custom-regression workflow: enter all three exact conditions in one table and fit one shared model for the unknown parameters. Automatic Log Mode does not apply because the vertical shift is unknown.

Fit a shifted exponential model

Practice problem

The function $m$ is defined by

m(x)=p\,r^x+q,

where $p$ , $q$ , and $r$ are positive constants. The graph of $y=m(x)$ passes through $(0,5)$ , $(1,9)$ , and $(3,57)$ . Which choice gives the value of $m(2)$ ?

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Fit the supplied points with the stated shifted exponential form, then evaluate the requested input.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Read the exponent: In $c+ab^{(x-h)/k}$ , the base applies every $k$ input units.
Track the baseline: The base multiplies the distance from $c$ , not the entire output.
Convert the interval: The per-unit factor is $b^{1/k}$ .
Interpret the factor: Growth uses $100(r-1)\%$ ; decay uses $100(1-r)\%$ .
Rewrite carefully: Use exponent rules, then verify equivalent forms by graph overlap.
Solve the target: Graph an exponential expression against its target when direct algebra is not immediate.
Fit the table: Automatic Log Mode fits compatible positive exponential data of the form $ab^x$ . Zero or negative outputs and unknown vertical shifts trigger ordinary regression instead.
Choose the job: Evaluate a given model; regress only when parameters must be fitted.

Next lesson

Circle geometry with distance and midpoint

Turn coordinate relationships into circle centers, radii, equations, and exact areas.

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Free Practice

What you’ll learn

Prerequisites

Read the multiplier the exponent actually describes

Fast Desmos solution

Choose the job before choosing the tool

Work from a given model when…

Fit parameters from data when…

1. Read every layer of the model

2. Convert an interval factor into a per-unit factor

3. Rewrite a transformed exponent into standard form

4. Solve an exponential target by graphing both sides

Do not confuse a logarithmic graph with Log Mode

5. Use automatic Log Mode for exponential data

6. Example: Read the model, graph the target, and respect the domain

Read the transformed model

Graph the model and target

Apply the least-integer condition

Finish the solution

Turn the fitted base into a percent

First steps

Finish it

Practice problems

SAT practice problems

Interpret an interval factor

Rewrite a transformed exponential

Find the least allowable input

Fit a shifted exponential model

5 practice examples left

Quick recap

Circle geometry with distance and midpoint

Free question bank practice problems