GRE Quantitative Reasoning - Exponential Functions

Find one possible value of , given the following equation:

$343=7^{15t-12}$

Cannot be determined from the information given.

Explanation

We begin with the following:

$343=7^{15t-12}$

This can be rewritten as

$7^3=7^{15t-12}$

Recall that if you have two exponents with equal bases, you can simply set the exponents equal to eachother. Do so to get the following:

Solve this to get t.

Find one possible value of , given the following equation:

$343=7^{15t-12}$

Cannot be determined from the information given.

Explanation

We begin with the following:

$343=7^{15t-12}$

This can be rewritten as

$7^3=7^{15t-12}$

Recall that if you have two exponents with equal bases, you can simply set the exponents equal to eachother. Do so to get the following:

Solve this to get t.

Solve for .

$7^{2y}-3^{y+2}=0$

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

$y=\frac{2\ln3}{2 \ln 7+\ln3}$

Explanation

The first step is to make sure we don't have a zero on one side which we can easily take care of:

$7^{2y}=3^{y+2}$

Now we can take the logarithm of both sides using natural log:

$\ln 7^{2y}=\ln 3^{y+2}$

Note: we can apply the Power Rule here $\ln (x^y)=y\ln (x)$

$2y \ln 7=(y+2)\ln3$

$2y \ln 7=y\ln3+2\ln3$

$2y \ln 7-y\ln3=2\ln3$

$y(2 \ln 7-\ln3)=2\ln3$

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

Solve for :

$2^{x+1}=8^4$

Explanation

Step 1: Rewrite the right side as a power of :

$8^4=8\cdot8\cdot8\cdot8$

$8^4=2^3\cdot2^3\cdot2^3\cdot2^3$

$8^4=2^{3+3+3+3}=2^{3\cdot 4}=2^{12}$

Step 2: Rewrite the original equation:

$2^{x+1}=2^{12}$

Step 3: Since the bases are equal, I can set the exponents equal.

So, $x+1=12\implies x=11$

Solve for .

$7^{2y}-3^{y+2}=0$

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

$y=\frac{2\ln3}{2 \ln 7+\ln3}$

Explanation

The first step is to make sure we don't have a zero on one side which we can easily take care of:

$7^{2y}=3^{y+2}$

Now we can take the logarithm of both sides using natural log:

$\ln 7^{2y}=\ln 3^{y+2}$

Note: we can apply the Power Rule here $\ln (x^y)=y\ln (x)$

$2y \ln 7=(y+2)\ln3$

$2y \ln 7=y\ln3+2\ln3$

$2y \ln 7-y\ln3=2\ln3$

$y(2 \ln 7-\ln3)=2\ln3$

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

Solve for :

$2^{x+1}=8^4$

Explanation

Step 1: Rewrite the right side as a power of :

$8^4=8\cdot8\cdot8\cdot8$

$8^4=2^3\cdot2^3\cdot2^3\cdot2^3$

$8^4=2^{3+3+3+3}=2^{3\cdot 4}=2^{12}$

Step 2: Rewrite the original equation:

$2^{x+1}=2^{12}$

Step 3: Since the bases are equal, I can set the exponents equal.

So, $x+1=12\implies x=11$

The rate of growth of the Martian Transgalactic Constituency is proportional to the population. The population increased by 23 percent between 2530 and 2534 AD. What is the constant of proportionality?

Explanation

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where is an initial population value, and is the constant of proportionality.

Since the population increased by 23 percent between 2530 and 2534 AD, we can solve for this constant of proportionality:

$(1+0.23)y_0=y_0e^{k(2534-2530)}$

$1.23=e^{4k}$

$k=\frac{ln(1.23)}{4}=0.05$

The rate of growth of the bacteria in an agar dish is proportional to the population. The population increased by 150 percent between 1:15 and 2:30. What is the constant of proportionality?

Explanation

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where is an initial population value, and is the constant of proportionality.

Since the population increased by 150 percent between 1:15 and 2:30, we can solve for this constant of proportionality:

$(1+1.50)y_0=y_0e^{k(2:30 - 1:15)}$

Dealing in minutes:

$2.50=e^{75k}$

$k=\frac{ln(2.50)}{75}=0.012$

Explanation

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where is an initial population value, and is the constant of proportionality.

Since the population increased by 23 percent between 2530 and 2534 AD, we can solve for this constant of proportionality:

$(1+0.23)y_0=y_0e^{k(2534-2530)}$

$1.23=e^{4k}$

$k=\frac{ln(1.23)}{4}=0.05$

The rate of growth of the bacteria in an agar dish is proportional to the population. The population increased by 150 percent between 1:15 and 2:30. What is the constant of proportionality?

Explanation

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where is an initial population value, and is the constant of proportionality.

Since the population increased by 150 percent between 1:15 and 2:30, we can solve for this constant of proportionality:

$(1+1.50)y_0=y_0e^{k(2:30 - 1:15)}$

Dealing in minutes:

$2.50=e^{75k}$

$k=\frac{ln(2.50)}{75}=0.012$

Exponential Functions

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