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GRE Subject Test: Math : Logarithms

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Example Questions

Example Question #1 : Evaluating Logarithms

Solve: log_4 256=x

Possible Answers:

x=-4

x=10

x=6

x=4

Correct answer:

x=4

Explanation:

Step 1: Re-write the log equation as an exponential equation. To do this, take the base of the log function and raise it to the number on the right side of the equal sign. This new exponent is equal to the number to the right of the log base.

4^x=256

Step 2: Re-write the right hand side as a power of 4..

256=4^4

Step 3: Re-write the equation

4^x=4^4

Step 4: We have the same base, so we can equal the exponents..

x=4

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Example Question #1 : Logarithms

$Solve\ for\ x: log_{x}(49)=2$

Possible Answers:

$\pm7$

49^2

24.5

Correct answer:

$\pm7$

Explanation:

In order to solve for x, we must first rewrite the log in exponential form.

Every log is written in the below general form:

$log_{base}(result)=exponent\ is\ rewritten\ as: base^e^x^p^o^n^e^n^t=result$

In this case we have:

$log_{x}(49)=2$

This becomes:

x^2=49

We can solve this by taking the square root of both sides:

$\sqrt{x^2}=\sqrt{49}$

$x=\pm7$

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Example Question #2 : Logarithms

Solve for :

log_4 4096=x

Possible Answers:

x=7

x=4

x=9

x=2

x=6

Correct answer:

x=6

Explanation:

Use rules of logarithms...

Take the base of the log and raise it to the number on the right side of the equal sign (which becomes the exponent):

4^x=4096

$\\4^x=16^3 \\4^x=(4^2)^3 \\4^x=4^6 \\x=6$

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Example Question #1 : Logarithms

Evaluate: $\log_6 46656=x$

Possible Answers:

Correct answer:

Explanation:

Step 1: Write the expression in exponential form...

Given: $\log_a b=x\implies a^x=b$

$\log_6 46656=x\implies 6^x=46656$

Step 2: Convert the right hand side into a power of 6..

46656=6^6

Step 3: Re-write the equations...

6^x=6^6

Since the bases are equal, taking log of both sides will cancel them.

$\log_6 6^x=\log_6 6^6$

$\implies x=6$

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Example Question #4 : Logarithms

$Solve\ for\ x:\ log_{2}(8)=x$

Possible Answers:

Correct answer:

Explanation:

$In\ order\ to\ solve\ for\ x\ we\ must\ know\ how\ to\ rewrite\ a\ log\ in\ exponent\ form:$

$We\ can\ use\ the\ general\ rule:\ log_{b}(a)=x\\ when\ rewritten\ in\ exponential\ form\ we\ get: b^{x}=a$

$To\ Solve\ for\ x:\ log_{2}(8)=x;\ we\ rewrite\ as\ 2^{x}=8$

$We\ know\ that\ 2^{3}=8;\ so\ the\ value\ of\ x=3$

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Example Question #1 : Logarithmic Properties

Rewrite the following expression as a single logarithm

$5log_35x+log_3\frac{1}{4x^3}-log_3x^2$

Possible Answers:

3.076

6.06

$log_3\frac{3125x^3}{9}$

log_33125x

$log_3\frac{3125x^3}{4}$

Correct answer:

6.06

Explanation:

Recall a few properties of logarithms:

1.When adding logarithms of like base, we multiply the inside.

2.When subtracting logarithms of like base, we divide the inside.

3. When multiplying a logarithm by a number, we can raise the inside to that power.

So we begin with this:

$5log_35x+log_3\frac{1}{4x^3}-log_3x^2$

I would start with 3 to simplify the first log.

$log_3(5x)^5+log_3\frac{1}{4x^3}-log_3x^2$

$log_33125x^5+log_3\frac{1}{4x^3}-log_3x^2$

Next, use rule 1 on the first two logs.

$log_3\left(3125x^5\cdot\frac{1}{4x^3}\right)-log_3x^2$

$log_3\left(\frac{3125x^2}{4}\right)-log_3x^2$

Then, use rule 2 to combine these two.

$log_3\left(\frac{3125x^2}{4x^2}\right)=log_3\left(\frac{3125}{4}\right)\approx 6.06$

So our answer is 6.06.

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Example Question #2 : Logarithmic Properties

$Rewrite\ as\ one\ log: [log(x)+log(z)]-log(y)$

Possible Answers:

$log\left (\frac{xz}{y} \right )$

$log\left (\frac{x+z}{y} \right )$

$log\left (\frac{xy}{z} \right )$

$log\left ({xz}-{y} \right )$

Correct answer:

$log\left (\frac{xz}{y} \right )$

Explanation:

When combining logarithms into one log, we must remember that addition and multiplication are linked and subtraction and division are linked.

log(ab)=log(a)+log(b)

$log\left (\frac{a}{b} \right )=log(a)+log(b)$

In this case we have multiplication and division - so we assume anything that is negative, must be placed in the bottom of the fraction.

[log(x)+log(z)]-log(y)=log(xz)-log(y)

$log(xz)-log(y)=log\left ( \frac{xz}{y} \right )$

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Example Question #3 : Logarithmic Properties

$Rewrite\ in\ log\ form: 12^(^x^+^2^)=y$

Possible Answers:

$log_{y}(12)=x+2$

$log_{x+2}(y)=12$

$log_{12}(y)=x+2$

$log_{12}(x+2)=y$

Correct answer:

$log_{12}(y)=x+2$

Explanation:

When rewriting an exponential function as a log, we must follow the model below:

$log_{base}(result)=exponent$

A log is used to find an exponent. The above corresponds to the exponential form below:

base^(^e^x^p^o^n^e^n^t^)=result

$12^(^x^+^2^)=y\ rewritten\ in\ log\ form\ becomes:$

$log_{12}(y)=x+2$

$The\ base\ is\ 12,\ the\ exponent\ is\ (x+2)\ and\ the\ result\ is\ y.$

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Example Question #2 : Logarithmic Properties

$Rewrite\ in\ exponential\ form: log_{3}(22)=y$

Possible Answers:

3^y=22

y^3=22

22^y=3

3^2^2=y

Correct answer:

3^y=22

Explanation:

In order to rewrite a log, we must remember the pattern that they follow below:

$log_{base}(result)=exponent$

In this question we have:

$log_{3}(22)=y$

$The\ base\ is\ 3,\ the\ exponent\ is\ y\ and\ the\ result\ is\ 22.$

$We\ now\ rewrite\ this\ as:\ 3^y=22$

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Example Question #3 : Logarithms

Express $4\log_4 (x^2)-\log_4(y^{3})+\frac {1}{5}\log_4(z)$ as a single logarithm.

Possible Answers:

$\log_4 \bigg(\frac {x^8\sqrt[3]{z}}{y^5}\bigg)$

$\log_4 \bigg(\frac {x^5\sqrt[8]{z}}{y^3}\bigg)$

$\log_4 \bigg(\frac {x^8\sqrt[5]{z}}{y^3}\bigg)$

$\log_4 \bigg(\frac {x^8\sqrt[5]{z}}{y^{-3}}\bigg)$

Correct answer:

$\log_4 \bigg(\frac {x^8\sqrt[5]{z}}{y^3}\bigg)$

Explanation:

Step 1: Recall all logarithm rules:

$\log_a b+\log_a c=\log_a (bc)$

$\log_a b-\log_a c=log_a \bigg(\frac{b}{c}\bigg)$

$c\log_a b=\log_a (b^c)$

$\large a^{\frac {1}{b}}=\sqrt[b]{a}$

Step 2: Rewrite the first term in the expression..

$4\log_4 x^2=\log_4 (x^2)^4=log_4 (x^8)$

Step 3: Re-write the third term in the expression..

$\frac {1}{5}\log_4 (z)=\log_4 (z)^{\frac {1}{5}}=log_4 (\sqrt[5] {z})$

Step 4: Add up the positive terms...

$\log_4(x^8)+\log_4 (\sqrt[5]{z})=\log_4 (x^8\sqrt[5]{z})$

Step 5: Subtract the answer the other term from the answer in Step 4.

$\log_4 (x^8\sqrt[5]{z})-\log_4(y^3)=\log_4 \bigg(\frac {x^8\sqrt[5]{z}}{y^3}\bigg)$

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GRE Subject Test: Math : Logarithms

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Evaluating Logarithms

Example Question #1 : Logarithms

Example Question #2 : Logarithms

Example Question #1 : Logarithms

Example Question #4 : Logarithms

Example Question #1 : Logarithmic Properties

Example Question #2 : Logarithmic Properties

Example Question #3 : Logarithmic Properties

Example Question #2 : Logarithmic Properties

Example Question #3 : Logarithms

All GRE Subject Test: Math Resources

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