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GRE Subject Test: Math : Logarithmic Properties

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Example Questions

Example Question #1 : Logarithms

Rewrite the following expression as a single logarithm

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

Possible Answers:

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

log_33125x

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

6.06

3.076

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 #1 : Logarithms

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

Possible Answers:

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

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

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

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

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 #1 : Logarithmic Properties

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

Possible Answers:

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

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

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

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

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 #122 : Algebra

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

Possible Answers:

3^y=22

22^y=3

y^3=22

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 #121 : Algebra

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

Possible Answers:

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

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

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

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

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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Example Question #2 : Logarithmic Properties

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

Possible Answers:

log(x)+log(y)-log(z)

log(x)-log(y)+log(z)

$\frac{log(x)*log(y)}{log(z)}$

log(x)*log(y)-log(z)

Correct answer:

log(x)+log(y)-log(z)

Explanation:

In order to expand this log, we must remember the log rules.

$Similar\ to\ the\ exponent\ rules\ when\ variables\ are\ multiplied\ within\ a\ log:\\ the\ variables\ are\ added: log(xy)=log(x)+log(y)$

$When\ variables\ are\ divided\ within\ a\ log\ they\ are\ subtracted:\\ log\left ( \frac{x}{y} \right )=log(x)-log(y)$

$For\ this\ problem\ we\ combine\ both\ rules:$

$Expand: log\left ( \frac{xy}{z} \right )=log(x)+log(y)-log(z)$

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Example Question #1 : Logarithmic Properties

$Solve\ for\ x: log_{3}(x)=2$

Possible Answers:

$\frac{1}{3}$

Correct answer:

Explanation:

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

$In\ this\ case\ we\ have\ log_{3}(x)=2$

$This\ can\ be\ rewritten\ in\ exponential\ form\ as: 3^2=x$

$We\ know\ that\ 3^2=9;\ so\ x=9$

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Example Question #3 : Logarithmic Properties

$Rewrite\ as\ one\ log:\ log(a)-(\left log(b)+log(c) \right)$

Possible Answers:

log(a-b-c)

log(a-b+c)

$log\left ( \frac{a}{bc} \right )$

$log\left ( \frac{ab}{c} \right )$

Correct answer:

$log\left ( \frac{a}{bc} \right )$

Explanation:

$When\ condensing\ logs\ we\ must\ remember\ the\ log\ properties:$

$log(x)+log(y)=log(xy): When\ adding;\ we\ multiply.$

$log(x)-log(y)=log\left (\frac{x}{y} \right ): When\ subtracting;\ we\ divide.$

$GIVEN: Rewrite\ as\ one\ log:\ log(a)-(\left log(b)+log(c) \right)\\$

$This\ is\ rewritten\ as:\ log \left (\frac{a}{bc} \right )\\ because\ the\ negative\ will\ be\ distributed\ to\ log(b)\ and\ log(c)$

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Example Question #2 : Logarithmic Properties

Rewrite the following expression as a single logarithm

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

Possible Answers:

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

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

6.06

3.076

log_33125x

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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GRE Subject Test: Math : Logarithmic Properties

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

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Logarithms

Example Question #1 : Logarithms

Example Question #1 : Logarithmic Properties

Example Question #122 : Algebra

Example Question #121 : Algebra

Example Question #2 : Logarithmic Properties

Example Question #1 : Logarithmic Properties

Example Question #3 : Logarithmic Properties

Example Question #2 : Logarithmic Properties

All GRE Subject Test: Math Resources

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