Multiplying and dividing Logarithms - 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 .

Answer

Using properties of logs we get:

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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:

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:

The logarithm can be converted to exponential form:

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

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

$ln $\left($\frac{x^{3}$$$y^{-2}$$}{2z^{4}$}\right)$

Answer

By logarithmic properties:

;

Combining these three terms gives the correct answer:

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Question

Many textbooks use the following convention for logarithms:

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

Solve:

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

Answer

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 , which is .

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Question

Find the value of the Logarithmic Expression.

Answer

Use the change of base formula to solve this equation.

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

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Question

Which of the following is equivalent to

$\log(4000)$ ?

Answer

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

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

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Question

What is another way of expressing the following?

$\small 3\log(2)$

Answer

Use the rule

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

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

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Question

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

Answer

In order to solve this problem you must understand the product property of logarithms 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:

now use the power property to move the exponent over:

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Question

Which of the following is equivalent to ?

Answer

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

This means that:

Split up these logarithms by addition.

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

The answer is:

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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}$)$

Compare your answer with the correct one above

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 .

Answer

Using properties of logs we get:

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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:

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:

The logarithm can be converted to exponential form:

Factor the equation:

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

Compare your answer with the correct one above

Question

Rewrite the following logarithmic expression in expanded form (i.e. as a sum and/or difference):

$ln $\left($\frac{x^{3}$$$y^{-2}$$}{2z^{4}$}\right)$

Answer

By logarithmic properties:

;

Combining these three terms gives the correct answer:

Compare your answer with the correct one above

Question

Many textbooks use the following convention for logarithms:

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

Solve:

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

Answer

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 , which is .

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Question

Find the value of the Logarithmic Expression.

Answer

Use the change of base formula to solve this equation.

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

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