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Algebra II : Logarithms

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Example Questions

Example Question #14 : Logarithms With Exponents

Solve: $log_{4}(8^3)$

Possible Answers:

$\frac{9}{2}$

$\frac{8}{3}$

Correct answer:

$\frac{9}{2}$

Explanation:

Rewrite the log so that the terms are in a fraction.

$log_{4}(8^3) = \frac{log(8^3)}{log( 4)}$

Both terms can now be rewritten in base two.

$\frac{log(8^3)}{log( 4)} = \frac{log((2^3)^3)}{log( 2^2)} = \frac{log(2)^9}{log(2)^2}$

The exponents can be moved to the front as coefficients.

$\frac{log(2)^9}{log(2)^2} = \frac{9log(2)}{2log(2)} = \frac{9}{2}$

The answer is: $\frac{9}{2}$

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Example Question #73 : Simplifying Logarithms

Which statement is true of $\frac{1}{5} \log _{5}N$ for all positive values of ?

Possible Answers:

$\frac{1}{5} \log _{5}N = \log_{5} \sqrt[5]{N}$

$\frac{1}{5} \log _{5}N = N$

$\frac{1}{5} \log _{5}N = \log_{5} (N - 1)$

$\frac{1}{5} \log _{5}N = \log_{5} (N - 5)$

$\frac{1}{5} \log _{5}N = \log_{5}\left ( \frac{1}{5} N \right )$

Correct answer:

$\frac{1}{5} \log _{5}N = \log_{5} \sqrt[5]{N}$

Explanation:

By the Logarithm of a Power Property, for all real , all N > 0 , $a > 0 , a \ne 1$

$b \log_{a} N = \log_{a}( N ^{b})$

Setting $b = \frac{1}{5} , a = 5$ , the above becomes

$\frac{1}{5} \log _{5}N = \log_{5} \left ( N^{\frac{1}{5}}\right )$

Since, for any a,b for which the expressions are defined,

$N^{\frac{1}{b}} = \sqrt[b]{N}$ ,

setting b = 5 , th equation becomes

$\frac{1}{5} \log _{5}N = \log_{5} \sqrt[5]{N}$ .

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Example Question #15 : Logarithms With Exponents

Which statement is true of

$\log_{7} \sqrt[N]{7}$

for all integers N > 1 ?

Possible Answers:

$\log_{7} \sqrt[N]{7}= \frac{1}{N}$

$\log_{7} \sqrt[N]{7} = N - 1$

$\log_{7} \sqrt[N]{7}= \frac{7}{N}$

$\log_{7} \sqrt[N]{7} = -N$

$\log_{7} \sqrt[N]{7} = N - 7$

Correct answer:

$\log_{7} \sqrt[N]{7}= \frac{1}{N}$

Explanation:

Due to the following relationship:

$\sqrt[N]{7} = 7 ^{\frac{1}{N}}$ ; therefore, the expression

$\log_{7} \sqrt[N]{7}$

can be rewritten as

$\log_{7} \left ( 7 ^{\frac{1}{N}} \right )$

By definition,

$\log_{a} \left (a^{M} \right )= M$ .

Set a = 7 and $M = \frac{1}{N}$ , and the equation above can be rewritten as

$\log_{7} \left ( 7 ^{\frac{1}{N}} \right )= \frac{1}{N}$ ,

or, substituting back,

$\log_{7} \sqrt[N]{7}= \frac{1}{N}$

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Example Question #1 : Solving Logarithmic Functions

Solve for :

$5^{x} = \left(\frac{1}{25}\right)^{4x-1}$

Possible Answers:

$x = \frac{1}{3}$

$x = \frac{1}{2}$

$x = \frac{2}{9}$

$x = \frac{-2}{7}$

$x = \frac{-1}{2}$

Correct answer:

$x = \frac{2}{9}$

Explanation:

To solve for , first convert both sides to the same base:

$5^{x} = \frac{1}{25}^{4x-1} = (5^{-2})^{4x-1} = 5^{-8x+2}$

Now, with the same base, the exponents can be set equal to each other:

x= - 8x +2

Solving for gives:

$x = \frac{2}{9}$

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Example Question #2 : Solving Logarithms

Solve the equation:

$\log{_{4}}{x^{2}}+\log{_{4}}{3}=1$

Possible Answers:

$\frac{\sqrt{6}}{3}, -\frac{\sqrt{6}}{3}$

$\frac{\sqrt{3}}{3}, -\frac{\sqrt{3}}{3}$

$\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}$

1,-1

$\frac{2\sqrt{3}}{3}, -\frac{2\sqrt{3}}{3}$

Correct answer:

$\frac{2\sqrt{3}}{3}, -\frac{2\sqrt{3}}{3}$

Explanation:

$\log{_{4}}{x^{2}}+\log{_{4}}{3}=1$

$\Rightarrow \log{_{4}}{x^{2}}+\log{_{4}}{3}=\log{_{4}}{4}$

$\Rightarrow log{_{4}}{(x^{2}\times 3)}=\log{_{4}}{4}$

$\Rightarrow 3x^{2}=4$

$\Rightarrow x^{2}=\frac{4}{3}$

$\Rightarrow x=\frac{2\sqrt{3}}{3}, x=-\frac{2\sqrt{3}}{3}$

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Example Question #3 : Solving Logarithms

Use $\small \small \log_23\approx 1.5850$ to approximate the value of $\small \log_2 24$ .

Possible Answers:

$\small \log_2 24\approx 3.4150$

$\small \log_2 24\approx 12.5850$

$\small \log_2 24\approx 3.5850$

$\small \log_2 24\approx 4.5850$

$\small \log_2 24\approx 12.6800$

Correct answer:

$\small \log_2 24\approx 4.5850$

Explanation:

Rewrite as a product that includes the number :

$\small \log_2 24 = \log_2 (2^3 \cdot 3)$

Then we can split up the logarithm using the Product Property of Logarithms:

$\small \log_2 (2^3\cdot 3) = \log_2 2^3 + \log_2 3$

$\small =3+\log_2 3$

$\small \approx 3 + 1.5850$

Thus,

$\small \log_2 24 \approx 4.5850$ .

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Example Question #3 : Solving Logarithms

Solve for .

$log_{9}^{ }(2x+1)=2$

Possible Answers:

x=40

$x=\frac{17}{2}$

$x=\frac{81}{2}$

x=20

x=41

Correct answer:

x=40

Explanation:

Rewrite in exponential form:

$9^{2}=2x+1$

Solve for x:

81=2x+1

x=40

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Example Question #4 : Solving Logarithms

Solve the following equation:

ln(3x+8)=0

Possible Answers:

$-\frac{8}{3}$

$\frac{8}{7}$

$-\frac{7}{3}$

$\frac{1}{3}$

Correct answer:

$-\frac{7}{3}$

Explanation:

For this problem it is helpful to remember that,

ln(3x+8) = 0 is equivalent to ln(3x+8) = ln(1) because ln(1)=0

Therefore we can set what is inside of the parentheses equal to each other and solve for as follows:

3x+8=1

3x=-7

$x=-\frac{7}{3}$

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Example Question #5 : Solving Logarithms

Solve for :

log_6x=5

Possible Answers:

15625

7776

10000

Correct answer:

7776

Explanation:

To solve this logarithm, we need to know how to read a logarithm. A logarithm is the inverse of an exponential function. If a exponential equation is

a^c = b

then its inverse function, or logarithm, is

log_ab=c

Therefore, for this problem, in order to solve for , we simply need to solve

6^5=x

which is 7776 .

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Example Question #7 : Solving Logarithms

$\log x+\log\left ( x+21\right )=2$

Solve for .

Possible Answers:

100

Correct answer:

Explanation:

Logs are exponential functions using base 10 and a property is that you can combine added logs by multiplying.

$x(x+21)=10^{2}$

$x^{2}+21x=100$

(x+25)(x-4)=0

You cannot take the log of a negative number. x=-25 is extraneous.

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Algebra II : Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #14 : Logarithms With Exponents

Example Question #73 : Simplifying Logarithms

Example Question #15 : Logarithms With Exponents

Example Question #1 : Solving Logarithmic Functions

Example Question #2 : Solving Logarithms

Example Question #3 : Solving Logarithms

Example Question #3 : Solving Logarithms

Example Question #4 : Solving Logarithms

Example Question #5 : Solving Logarithms

Example Question #7 : Solving Logarithms

All Algebra II Resources

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