Logarithms

Help Questions

Algebra II › Logarithms

Questions 1 - 10

Solve .

$\frac{e^{2}}{3}$

$\frac{e^3}{2}$

$e^{2}$

$\frac{e}{2}$

Explanation

First we start by subtracting from each side:

Next, we rewrite the equation in exponent form:

Finally, we divide by :

$x=\frac{e^2}{3}$

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.

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

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

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

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.

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

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

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

Solve the following:

$\log10000+\log10$

$\log10010$

Explanation

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\log 10000=x$

is asking:

For the second term,

$\log(10)=y$

is asking:

So, our final answer is

Solve: $2log_{10}(100^3)$

$\textup{Cannot be determined.}$

Explanation

Change the base of the inner term or log to base ten.

$2log_{10}(100^3)= 2log_{10}(10^{2(3)})= 2log_{10}(10^6)$

According to the log property:

$log_{b}(b^x) = x$

The log based ten and the ten to the power of will cancel, leaving just the power.

$2log_{10}(10^6) = 2(6) =12$

The answer is:

Solve the following:

$\log10000+\log10$

$\log10010$

Explanation

When the base isn't explicitly defined, the log is base 10. For our problem, the first term

$\log 10000=x$

is asking:

For the second term,

$\log(10)=y$

is asking:

So, our final answer is

Solve: $2log_{10}(100^3)$

$\textup{Cannot be determined.}$

Explanation

Change the base of the inner term or log to base ten.

$2log_{10}(100^3)= 2log_{10}(10^{2(3)})= 2log_{10}(10^6)$

According to the log property:

$log_{b}(b^x) = x$

The log based ten and the ten to the power of will cancel, leaving just the power.

$2log_{10}(10^6) = 2(6) =12$

The answer is:

Which is another way of expressing

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

$\small \log(20)$

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

$\small \log(9)$

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

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

Explanation

Use the rule:

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

therefore

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

Add the logarithms:

$log_{10}(2)+log_{10}(7)$

$log_{10}(14)$

$log_{10}(9)$

$log_{10}(4)$

$log_{10}(12)$

Explanation

When adding logarithms of the same base, all you have to do is multiply the numbers inside the function as shown below:

$log_{10}(2)+log_{10}(7)$

$log_{10}(2*7)$

$log_{10}(14)$

Add the logarithms:

$log_{10}(2)+log_{10}(7)$

$log_{10}(14)$

$log_{10}(9)$

$log_{10}(4)$

$log_{10}(12)$

Explanation

When adding logarithms of the same base, all you have to do is multiply the numbers inside the function as shown below:

$log_{10}(2)+log_{10}(7)$

$log_{10}(2*7)$

$log_{10}(14)$

Page 1 of 69

Return to subject