Simplifying Logarithms - Math

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Question

Simplify the expression using logarithmic identities.

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Answer

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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Question

Which of the following represents a simplified form of ?

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Answer

The rule for the addition of logarithms is as follows:

.

As an application of this,.

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Question

Simplify $log_{5}75 - log_{5}3$ .

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Answer

Using properties of logs we get:

$log_{5}75 - log_{5}3 = log_{5}\frac{75}{3}= log_{5}25$

$log_{5}25= log_{5}5^{2}= 2log_{5}5= 2\cdot 1=2$

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Question

Simplify the following expression:

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Answer

Recall the log rule:

In this particular case, and . Thus, our answer is $log(4\cdot 12) = log(48)$ .

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Question

Use the properties of logarithms to solve the following equation:

$log_{6}(n-3)+log_{6}(n+2)=log_{3}(3)$

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Answer

Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:

$log_{6}(n-3)+log_{6}(n+2)=log_{3}(3)$

$log_{6}(n-3)(n+2)=1$

The logarithm can be converted to exponential form:

$(n-3)(n+2)=6^{1}$

Factor the equation:

Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .

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Question

Evaluate by hand $\log_2(4^{\log_2(4^{5})})$

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Answer

Using the logarithm rules, exponents within logarithms can be removed and simply multiplied by the remaining logarithm. This expression can be simplified as $5\log_2(4)\log_2(4)$

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Question

Solve for

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Answer

Use the power reducing theorem:

$\log a^x = b \rightarrow x\log a = b$

and

$x\log 2 = 6$

$x= \frac{6}{\log 2}$

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Question

Which of the following expressions is equivalent to ?

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Answer

According to the rule for exponents of logarithms,. As a direct application of this,.

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Question

Simplify the expression below.

$log(2 \cdot 2 \cdot 2)$

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Answer

Based on the definition of exponents, $log(2 \cdot 2 \cdot 2) = log (2^{3})$ .

Then, we use the following rule of logarithms:

$log(m^{n}) = n\cdot log(m)$

Thus, $log(2^{3}) = 3log(2)$ .

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Question

Simplify the expression using logarithmic identities.

Tap to reveal answer

Answer

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.

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

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

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

← Didn't Know|Knew It →

Question

Which of the following represents a simplified form of ?

Tap to reveal answer

Answer

The rule for the addition of logarithms is as follows:

.

As an application of this,.

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Question

Simplify $log_{5}75 - log_{5}3$ .

Tap to reveal answer

Answer

Using properties of logs we get:

$log_{5}75 - log_{5}3 = log_{5}\frac{75}{3}= log_{5}25$

$log_{5}25= log_{5}5^{2}= 2log_{5}5= 2\cdot 1=2$

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Question

Simplify the following expression:

Tap to reveal answer

Answer

Recall the log rule:

In this particular case, and . Thus, our answer is $log(4\cdot 12) = log(48)$ .

← Didn't Know|Knew It →

Question

Use the properties of logarithms to solve the following equation:

$log_{6}(n-3)+log_{6}(n+2)=log_{3}(3)$

Tap to reveal answer

Answer

Since the bases of the logs are the same and the logarithms are added, the arguments can be multiplied together. We then simplify the right side of the equation:

$log_{6}(n-3)+log_{6}(n+2)=log_{3}(3)$

$log_{6}(n-3)(n+2)=1$

The logarithm can be converted to exponential form:

$(n-3)(n+2)=6^{1}$

Factor the equation:

Although there are two solutions to the equation, logarithms cannot be negative. Therefore, the only real solution is .

← Didn't Know|Knew It →

Question

Evaluate by hand $\log_2(4^{\log_2(4^{5})})$

Tap to reveal answer

Answer

Using the logarithm rules, exponents within logarithms can be removed and simply multiplied by the remaining logarithm. This expression can be simplified as $5\log_2(4)\log_2(4)$

← Didn't Know|Knew It →

Question

Solve for

Tap to reveal answer

Answer

Use the power reducing theorem:

$\log a^x = b \rightarrow x\log a = b$

and

$x\log 2 = 6$

$x= \frac{6}{\log 2}$

← Didn't Know|Knew It →

Question

Which of the following expressions is equivalent to ?

Tap to reveal answer

Answer

According to the rule for exponents of logarithms,. As a direct application of this,.

← Didn't Know|Knew It →

Question

Simplify the expression below.

$log(2 \cdot 2 \cdot 2)$

Tap to reveal answer

Answer

Based on the definition of exponents, $log(2 \cdot 2 \cdot 2) = log (2^{3})$ .

Then, we use the following rule of logarithms:

$log(m^{n}) = n\cdot log(m)$

Thus, $log(2^{3}) = 3log(2)$ .

← Didn't Know|Knew It →

Question

Simplify the expression using logarithmic identities.

Tap to reveal answer

Answer

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.

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

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

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

← Didn't Know|Knew It →

Question

Which of the following represents a simplified form of ?

Tap to reveal answer

Answer

The rule for the addition of logarithms is as follows:

.

As an application of this,.

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