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Algebra II : Adding and Subtracting Logarithms

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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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Example Question #5 : Adding And Subtracting Logarithms

What is the value of (log_7343 + log_749)-log_72401 ?

Possible Answers:

Correct answer:

Explanation:

Remember the rules of logarithms:

log_ab + log_ac = log_abc

$log_pq - log_pr = log_p\frac{q}{r}$

This means we can simplify it as follows:

(log_7343 + log_749)-log_72401

$log_7(343\cdot 49) - log_72401$

log_716807 - log_72401

$log_7\frac{16807}{2401}$

log_77

The logarithm of anything with the same base is always , so the correct answer is .

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Example Question #1 : Adding And Subtracting Logarithms

Which of the following is another way to express

$\small \log (8)-\log(4)$ ?

Possible Answers:

$\small \log(12)$

$\small \log (6)$

$\small \log(\frac{1}{2})$

$\small \log(4)$

$\small \log (2)$

Correct answer:

$\small \log (2)$

Explanation:

Use the rule

$\small \log(a)-\log (b)=\log(\frac{a}{b})$

therefore

$\small \log (8)-\log(4)=\log(\frac{8}{4})=\log (2)$

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Example Question #7 : Adding And Subtracting Logarithms

Which is another way of expressing

$\small \log(4)+\log(5)$ ?

Possible Answers:

$\small \small \frac{\log (4)}{\log (5)}$

$\small \log(20)$

$\small \log(9)$

$\small \log(\frac{4}{5})$

$\small \log _{4}(5)$

Correct answer:

$\small \log(20)$

Explanation:

Use the rule:

$\small \log(a)+\log(b)=\log(a\times b)$

therefore

$\small \log (4)+\log (5)=\log (4\times 5)=\log (20)$

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Example Question #8 : Adding And Subtracting Logarithms

Simplify

log_5x-log_52y

Possible Answers:

log_5x2y

2ylog_5x

$log_5\frac{x}{2y}$

$log_5\frac{2y}{x}$

Correct answer:

$log_5\frac{x}{2y}$

Explanation:

This problem can be solved using the properties of logs. When two logs are being subtracted from each other, it is the same thing as dividing two logs together. Remember that to use this rule, the logs must have the same base in this case log_5 .

$log_5x-log_52y=log_5\frac{x}{2y}$

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Example Question #9 : Adding And Subtracting Logarithms

Condense this logarithm: $\frac{1}{2}\log_{10}x+3\log_{10}(x+1)$

Possible Answers:

$\log_{10}\left( \sqrt{x}+\sqrt{x+1}\right)$

$\frac{1}{2}\log_{10}\left[x+\log_{10}(x+1)^3\right]$

$\log_{10}\left[\sqrt{x}+3\log_{10}(x+1)\right]$

$\log_{10}\left[\sqrt{x}+\log_{10}(x+1)^3\right]$

$\log_{10}\left[\sqrt{x}(x+1)^3\right]$

Correct answer:

$\log_{10}\left[\sqrt{x}(x+1)^3\right]$

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.

$\frac{1}{2}\log_{10}x+3\log_{10}(x+1)$

first move the constants in front of the logarithmic functions to their proper place using the power rule.

$\log_{10}x^{\frac{1}{2}}+\log_{10}(x+1)^3$

next factor out the logarithmic equation:

$\log_{10}[x^\frac{1}{2}+(x+1)^3]$

change the fractional exponent to a radical

$\log_{10}[\sqrt{x}(x+1)^3]$

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Example Question #10 : Adding And Subtracting Logarithms

Expand $log_{8}(\frac{15}{8})$

Possible Answers:

$log_8(15)\ast log_8(8)$

$log_{8}(15)-log_{8}(8)$

$log_{8}(15)+log_8(8)$

$\frac{log_8(15)}{log_8(8)}$

Correct answer:

$log_{8}(15)-log_{8}(8)$

Explanation:

The rule for expanding and dividing logarithms is that you can subtract the terms inside the log. In this case, the question is not asking for an actual number, but just what the expanded version would be. Therefore you separate the terms inside the log by subtracting the denominator from the numerator. Therefore the answer is

$log_{8}(15)-log_{8}(8)$

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Algebra II : Adding and Subtracting Logarithms

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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

Example Question #5 : Adding And Subtracting Logarithms

Example Question #1 : Adding And Subtracting Logarithms

Example Question #7 : Adding And Subtracting Logarithms

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