Logarithm Base

Identify the base in the following.

1. ${\log }_{3}9=2$
2. $5^4=625$

Solution

1. The base here is 3. It is the subscript of the logarithm symbol log.
2. The base here is 5. It is the number that raises the exponent 4.

Give the answers to the following.

1. $\log \left(1000\right)$
2. $\log \left(20\right)$
3. $\log \left(8\right)$

This means that if you multiplying 10 in three places gives 1000 that is ${10}^3$.

Give the answers to the following

a. ${\log }_{e}7.3$

b. $\ln \left(25\right)$

c. ${\log }_{e}33.98$

Solution

1. To get the answer to ${\log }_{e}7.3$, you'll need a calculator. You will need to press the $\ln$ button on the calculator and then 7.3. The answer will appear after that.

${\log }_{e}7.3=1.9878$

b. Using a calculator,$\ln \left(25\right)=3.2188$

c. Using a calculator,${\log }_e\left(33.98\right)=3.5257$

Simplify $y={\log }_2\left(20\right)$

Solution

The first thing is to change the base using the change of base formula. You can change the base to any number including base 10 and the natural logarithm e. You just have to make sure that they are both the same base. Doing this we will have:

${\log }_2\left(20\right)=\frac{{\log }_{10}\left(20\right)}{{\log }_{10}\left(2\right)}$

We will use a calculator to solve the numerator and denominator to get:

$$\begin{gathered} \frac{{\log }_{10}\left(20\right)}{{\log }_{10}\left(2\right)}=\frac{1.3010}{0.3010} \\ =4.32 \end{gathered}$$

Solve ${\log }_3\left(x\right)={\log }_9\left(4\right)$

Solution

You will notice that there are different bases involved so we will use the change of base formula. We can change both base to 3 or 9 and you will still arrive at the same answer. Remember the aim is just to make sure that both bases are equal.

We will use the change of base formula on the right hand side. This means we are making the bases 3.

$$\begin{gathered} {\log }_3\left(x\right)=\frac{{\log }_3\left(4\right)}{{\log }_3\left(9\right)} \\ {\log }_3\left(x\right)=\frac{{\log }_3\left(4\right)}{{\log }_3\left(3^2\right)} \end{gathered}$$

There is a logarithm law in the form ${\log }_b\left(b^n\right)=n$. We will apply this law to the denominator present and we will have:

$$\begin{gathered} {\log }_b\left(b^n\right)=n \\ {\log }_3{\left(3\right)}^2=2 \end{gathered}$$

We will put the result "2" in the equation and continue solving.

$$\begin{gathered} {\log }_3\left(x\right)=\frac{{\log }_3\left(4\right)}{2} \\ {\log }_3\left(x\right)=\frac{1}{2}{\log }_3\left(4\right) \end{gathered}$$

There is another logarithm law in the form ${\log }_a\left(x^n\right)=n{\log }_a\left(x\right)$. If we apply this we will get:

$$\begin{gathered} {\log }_3\left(x\right)={\log }_3\left(4^{\frac{1}{2}}\right) \\ {\log }_3\left(x\right)={\log }_3\left(2\right) \end{gathered}$$

Using the rule that if ${\log }_b\left(x\right)={\log }_b\left(y\right)$, then $x=y$, our final answer will be:

$x=2$

Solve ${\log }_9\left(4\right)+{\log }_3\left(x\right)=3$

Solution

The first thing is to make the bases the same. We can choose to make them both 9 or 3. Either way, we will arrive at the same answer. Let's make them both 3.

The change of base formula is: ${\log }_b\left(x\right)=\frac{{\log }_a\left(x\right)}{{\log }_a\left(b\right)}$

We will use the change of base formula on the first term of the expression and we will get:

$\frac{{\log }_3\left(4\right)}{{\log }_3\left(9\right)}+{\log }_3\left(x\right)=3$

If you observe, you will see that you can simplify the denominator with a calculator or manually. You can tell that the result is 2 because 3 squared is 9. So, we will now have:

$\frac{{\log }_3\left(4\right)}{2}+{\log }_3\left(x\right)=3$

Let's multiply each term by 2

${\log }_3\left(4\right)+2{\log }_3\left(x\right)=6$

We can use the power logarithmic rule on the second expression which is ${\log }_b\left(x^n\right)=n{\log }_b\left(x\right)$

We will now have ${\log }_3\left(4\right)+{\log }_3\left(x^2\right)=6$

We can use the addition rule here which is ${\log }_b\left(x\right)+{\log }_b\left(y\right)={\log }_b\left(xy\right)$

Therefore ${\log }_3\left(4x^2\right)=6$

We will now take the anti log to get

$4x^2=3^6$

What we simplify did here was to raise the base 3 to the power of 6.

The next and final step is to find x

$x^2=\frac{3^6}{4}$

Take the square root of both sides

$$\begin{gathered} \sqrt{x^2}=\sqrt{\frac{3^6}{4}} \\ \cancel{\sqrt{x{}^{\cancel{2}}}}=\frac{27}{2} \\ x=\frac{27}{2}=13.5 \end{gathered}$$

Plot the logarithmic expressions and observe the plot.

$y={\log }_2\left(x\right)$ and $y={\log }_{10}\left(x\right)$

Solution

What you need to do is make a table for both expressions and plot the graph.

For $y={\log }_2\left(x\right)$

For $y={\log }_{10}\left(x\right)$

We will now plot the graph

You can see that $y={\log }_{10}\left(x\right)$ is closer to the y axis.