The function is defined by For every increase of in the value of , the value of increases by a factor of , where is a constant. Which choice rewrites so that the value of appears as the base of the exponential expression?

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Super-Hard SAT Math Question 56 | Free SAT Prep | Aniko

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Question 56·200 Super-Hard SAT Math Questions·Advanced Math

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The function $g$ is defined by

g(x)=10(12)^{x/2}(18)^{x/3}.

For every increase of $\frac{2}{3}$ in the value of $x$ , the value of $g(x)$ increases by a factor of $c$ , where $c$ is a constant.

Which choice rewrites $g(x)$ so that the value of $c$ appears as the base of the exponential expression?

Convert all exponential bases to prime powers so you can combine them into a single expression of the form $A\cdot B^{kx}$ . Then use $\dfrac{g(x+\Delta x)}{g(x)}=B^{k\Delta x}$ to identify the constant factor for that step size, and rewrite $B^{kx}$ as a power of that factor so the base equals $c$ .

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Factor the bases

Rewrite $12$ and $18$ using prime factors ( $2$ and $3$ ), then apply $(ab)^k=a^k b^k$ .

Combine like bases

After rewriting, combine the powers of $2$ together and the powers of $3$ together by adding exponents.

Use a ratio for the constant factor

Translate “increases by a factor of $c$ when $x$ increases by $\tfrac{2}{3}$ ” into $\dfrac{g\left(x+\tfrac{2}{3}\right)}{g(x)}$ and simplify.

Desmos Guide

Enter the original function

In Desmos, enter: g(x)=10*(12)^(x/2)*(18)^(x/3).

Check the factor for a step of $2/3$

Enter r(x)=g(x+2/3)/g(x) and verify it stays constant as you change $x$ .

Verify the rewritten form

Enter h(x)=10*((2^8*3^7)^(1/9))^(3x/2) and confirm g(x) and h(x) overlap (or graph g(x)-h(x) and check it is $0$ ).

Step-by-step Explanation

Rewrite each factor using primes

Factor the bases:

12=2^2\cdot 3,\qquad 18=2\cdot 3^2.

(12)^{x/2}=(2^2\cdot 3)^{x/2}=2^{x}\cdot 3^{x/2}

and

(18)^{x/3}=(2\cdot 3^2)^{x/3}=2^{x/3}\cdot 3^{2x/3}.

Combine into one exponential base

Multiply the powers of 2 and 3:

2^{x}\cdot 2^{x/3}=2^{4x/3}, \qquad 3^{x/2}\cdot 3^{2x/3}=3^{x/2+2x/3}=3^{7x/6}.

Now write both with exponent $x/6$ :

2^{4x/3}=2^{8x/6}=(2^8)^{x/6}, \qquad 3^{7x/6}=(3^7)^{x/6}.

Therefore,

g(x)=10\,(2^8)^{x/6}(3^7)^{x/6}=10\left(2^8\cdot 3^7\right)^{x/6}.

Find the growth factor for a $\tfrac{2}{3}$ increase in $x$

A constant factor for an increase of $\tfrac{2}{3}$ means

\frac{g\left(x+\frac{2}{3}\right)}{g(x)} = \frac{10\left(2^8\cdot 3^7\right)^{(x+2/3)/6}}{10\left(2^8\cdot 3^7\right)^{x/6}} = \left(2^8\cdot 3^7\right)^{(2/3)/6} = \left(2^8\cdot 3^7\right)^{1/9}.

Rewrite $g(x)$ so the base is $c$

The ratio in the previous step is the constant factor $c$ , so

c=\left(2^8\cdot 3^7\right)^{1/9}.

Rewrite $\left(2^8\cdot 3^7\right)^{x/6}$ as a power of $c$ :

\left(2^8\cdot 3^7\right)^{x/6} = \left(\left(2^8\cdot 3^7\right)^{1/9}\right)^{\frac{x/6}{1/9}} = \left(\left(2^8\cdot 3^7\right)^{1/9}\right)^{3x/2}.

Therefore,

g(x)=10\left(\left(2^8\cdot 3^7\right)^{1/9}\right)^{3x/2}.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation