Simplifying Logarithms

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Questions 1 - 10

1

Simplify the expression using logarithmic identities.

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

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

The expression cannot be simplified

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.

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

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

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

2

Simplify the expression using logarithmic identities.

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

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

The expression cannot be simplified

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.

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

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

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

3

Solve for

Explanation

Use the power reducing theorem:

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

and

$x\log 2 = 6$

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

4

Solve for

Explanation

Use the power reducing theorem:

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

and

$x\log 2 = 6$

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

5

Which of the following expressions is equivalent to ?

Explanation

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

6

Which of the following expressions is equivalent to ?

Explanation

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

7

Which of the following represents a simplified form of ?

$\frac{log(5) }{ log(x) }$

Explanation

The rule for the addition of logarithms is as follows:

.

As an application of this,.

8

Which of the following represents a simplified form of ?

$\frac{log(5) }{ log(x) }$

Explanation

The rule for the addition of logarithms is as follows:

.

As an application of this,.

9

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

Explanation

Using properties of logs we get:

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

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

10

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

Explanation

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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