Simplifying Logarithms - High School Math

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Question

Simplify the expression using logarithmic identities.

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 ?

Answer

The rule for the addition of logarithms is as follows:

.

As an application of this,.

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Question

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

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:

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:

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

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 $\log_2(4^{\log_2(4^{5})})$

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

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 ?

Answer

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

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Question

Simplify the expression below.

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

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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