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Algebra II : Logarithms

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Example Question #2 : Multiplying And Dividing Logarithms

Which of the following is equivalent to

$\log(4000)$ ?

Possible Answers:

$10 + \log4$

2000

$3\log4$

$3 + \log4$

Correct answer:

$3 + \log4$

Explanation:

Recall that log implies base if not indicated.Then, we break up $4000 = 4\cdot 1000$ . Thus, we have $\log(4 \cdot 1000)$ .

Our log rules indicate that

$\log(ab) = \log(a) + \log(b)$ .

So we are really interested in,

$\log(4) + \log(1000)$ .

Since we are interested in log base , we can solve $\log(1000)$ without a calculator.

We know that 1000=10^3 , and thus by the definition of log we have that $\log(1000) = 3$ .

Therefore, we have $\log(4) + 3$ .

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Example Question #3 : Multiplying And Dividing Logarithms

Find the value of the Logarithmic Expression.

$\log_{3} \frac{1}{81}$

Possible Answers:

Correct answer:

Explanation:

Use the change of base formula to solve this equation.

$Log_{_{3}} \frac{1}{81}$

$y = Log_{3} \frac{1}{81}$

$3^{y} = \frac{1}{81}$

$3^{^{y}} = 3^{-4}$

y = -4

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Example Question #4 : Multiplying And Dividing Logarithms

What is another way of expressing the following?

$\small 3\log(2)$

Possible Answers:

$\small \small \log (\frac{2}{3})$

$\small \log (6)$

$\small \log (9)$

$\small \log (8)$

$\small \log (5)$

Correct answer:

$\small \log (8)$

Explanation:

Use the rule

$\small a\times \log (b)= \log (b^{a})$

$\small \log (2^{3})=\log (8)$

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Example Question #5 : Multiplying And Dividing Logarithms

Expand this logarithm: $\log_{10}8x^4y$

Possible Answers:

$\log_{10}8y+4\log_{10}x$

$\log_{10}y+\log_{10}x^4 +8$

$\log_{10}8+\log_{10}x^4+\log_{10}y$

$\log_{10}(8+y+x^4)$

$\log_{10}8+4\log_{10}x+\log_{10}y$

Correct answer:

$\log_{10}8+4\log_{10}x+\log_{10}y$

Explanation:

In order to solve this problem you must understand the product property of logarithms $\log_{10}uv=\log_{10}u+\log_{10}v$ and the power property of logarithms $\log_{10}u^n=n\log_{10}u$ . Note that these apply to logs of all bases not just base 10.

$\log_{10}8x^4y$

log of multiple terms is the log of each individual one:

$\log_{10}8+\log_{10}x^4+\log_{10}y$

now use the power property to move the exponent over:

$\log_{10}8+4\log_{10}x+\log_{10}y$

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Example Question #6 : Multiplying And Dividing Logarithms

Which of the following is equivalent to $log(\frac{a^2b^3}{x^{-3}})$ ?

Possible Answers:

2log(a)+log(b-x)^3

log(2a)+log(3b)+log(3x)

2log(a)+3log(b-x)

2log(a)+3log(b)-3log(x)

2log(a)+3log(b)+3log(x)

Correct answer:

2log(a)+3log(b)+3log(x)

Explanation:

We can rewrite the terms of the inner quantity. Change the negative exponent into a fraction.

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

$\frac{a^2b^3}{x^{-3}} = a^2b^3 \div \frac{1}{x^3}= a^2b^3 \times \frac{x^3}{1} =a^2b^3x^3$

This means that:

$log(\frac{a^2b^3}{x^{-3}}) = log(a^2b^3x^3)$

Split up these logarithms by addition.

log(a^2b^3x^3) = log(a^2)+log(b^3)+log(x^3)

According to the log rules, the powers can be transferred in front of the logs as coefficients.

The answer is: 2log(a)+3log(b)+3log(x)

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Example Question #7 : Multiplying And Dividing Logarithms

Many textbooks use the following convention for logarithms:

$log(x) = log_{10}(x)$

ln(x) = log_e(x)

lg(x)=log_2(x)

Solve:

$lg(\frac{4096}{16384})$

Possible Answers:

Correct answer:

Explanation:

Remembering the rules for logarithms, we know that $log_a(\frac{b}{c}) = log_ab - log_ac$ .

This tells us that $lg(\frac{4096}{16384})=lg(4096)-lg(16384)$ .

This becomes 12 - 14 , which is .

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Example Question #1 : Adding And Subtracting Logarithms

Simplify the following logarithmic expression:

$log_{3}81 - log_{5} (\frac{1}{25}) + log_{7} 7^{\frac{3}{2}}$

Possible Answers:

$\frac{21}{2}$

$\frac{7}{2}$

$\frac{15}{2}$

Correct answer:

$\frac{15}{2}$

Explanation:

Each term can be simplified as follows:

$log_{3}81 = log_{3} 3^{4} = 4$

$- log_{5} (\frac{1}{25}) = - log_{5} (5)^{-2} = -(-2) = 2$

$log_{7} 7^{\frac{3}{2}} = \frac{3}{2}$

Combining these gives the answer: $\frac{15}{2}$

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Example Question #2 : Adding And Subtracting Logarithms

Simplify the expression using logarithmic identities.

log_47-log_45

Possible Answers:

The expression cannot be simplified

$log_4(\frac{5}{7})$

log_4(35)

$log_4(\frac{7}{5})$

log_4(2)

Correct answer:

$log_4(\frac{7}{5})$

Explanation:

The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.

$log_A(\frac{B}{C})=log_AB-log_AC$

If we encounter two logarithms with the same base, we can likely combine them. In this case, we can use the reverse of the above identity.

$log_AB-log_AC=log_A(\frac{B}{C})$

$A=4,\ B=7,\ C=5$

$log_47-log_45=log_4(\frac{7}{5})$

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Example Question #3 : Adding And Subtracting Logarithms

Use logarithmic properties to simplify this expression:

$log(4x)+log\left(\frac{1}{2}\right)-log(y^{2})$

Possible Answers:

$log(2xy^{2})$

$log\left(\frac{2x^{2}}{y^{2}}\right)$

$log\left(\frac{2x}{y^{2}}\right)$

$log\left(\frac{8x}{y^{2}}\right)$

$log\left(\frac{4x}{y^{2}}\right)$

Correct answer:

$log\left(\frac{2x}{y^{2}}\right)$

Explanation:

Use the sum/product rule to combine the first 2 terms:

$log\left(4x\ast \frac{1}{2}\right)-log(y^{2})=log(2x)-log(y^{2})$

Use the difference/quotient rule to combine the remaining terms:

$log\left(\frac{2x}{y^{2}}\right)$

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Example Question #4 : Adding And Subtracting Logarithms

Expand the following logarithmic expression into a list of sums or subtractions of logarithms:

${}\log(\frac{a^3b^2}{c^4})$

Possible Answers:

${}3\log a - 2\log b - 4\log c$

${}6\log a + 6\log b - 4\log c$

${}3\log a -\log b -\log c$

${} 3\log a + 2\log b - 4\log c$

Correct answer:

${} 3\log a + 2\log b - 4\log c$

Explanation:

One important property of logarithms is that multiplication inside the logarithm is the same thing as addition outside of it. In the same way division is "the same" as subtraction in logarithms. So our expression is the same as

${} \log a^3 + \log b^2 - \log c^4$

But also, exponents can be moved outside in the same way. a^3 is basically $a \cdot a \cdot a$ , so log(a^3)=3log(a) . This can be reduced even further to our final answer:

$3 \log a+2 \log b-4 \log c$

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Algebra II : Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #2 : Multiplying And Dividing Logarithms

Example Question #3 : Multiplying And Dividing Logarithms

Example Question #4 : Multiplying And Dividing Logarithms

Example Question #5 : Multiplying And Dividing Logarithms

Example Question #6 : Multiplying And Dividing Logarithms

Example Question #7 : Multiplying And Dividing Logarithms

Example Question #1 : Adding And Subtracting Logarithms

Example Question #2 : Adding And Subtracting Logarithms

Example Question #3 : Adding And Subtracting Logarithms

Example Question #4 : Adding And Subtracting Logarithms

All Algebra II Resources

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