Question

How do you use the change of base formula and a calculator to evaluate the logarithm $\log_{4}24$ ?

Answer 1

In this question, we need to change the base of the given logarithmic expression $\log_{4}24$ by using a change of base formula. Then we have to find the value of the expression using a calculator. We cannot calculate $\log_{4}24$ directly using the calculator because the calculator doesn’t have a button named $\log_{4}$ . Therefore, first we need to change the base of the given expression . There is a direct formula which allows us to rewrite the given logarithmic expression with another base. With the use of the formula we can change the base of the given expression. Then we need to type the given function in the calculator. Thus by following the steps below we can find the value of $\log_{4}24$ using a calculator.

Formula used :
The change of base formula is used to write a logarithm of a number with a given base as the ratio of two logarithms each with the same base but which is different from the base of the original logarithm given.
$\log_{b}a\ = \dfrac{\left\lbrack \log_{c}a \right\rbrack}{\left\lbrack \log_{c}b \right\rbrack}$
Where $a$ , $b$ and $c$ are any positive real numbers.

Complete step by step answer:
Given, $\log_{4}24$
Here we need to change the base of the given expression.
$\log_{b}a\ = \dfrac{\left\lbrack \log_{c}a \right\rbrack}{\left\lbrack \log_{c}b \right\rbrack}$
On observing $\log_{4}24$ here $a = 24$ and $b = 4$
On applying in the change of base formula,
We get,
$\log_{4}24 = \dfrac{\log_{10}24}{\log_{10}4}$
Now we can use the calculator to find the logarithms values.
First we need to press the log button in our calculator. Then we need to put our input $24$ . We should write our input $24$ in brackets. For that we need to press the ( and ) symbol in our calculator. Then we need to press the divide symbol $( \div )$ in our calculator. Then again we need to press the log button then we need to put our input $4$ . We should write our input $4$ in brackets. For that we need to press the ( and ) symbol in our calculator. Then we need to press the equal to symbol $( = )$ to get our answer. Then the answer will be displayed on the display of the calculator.
We should get $2.29248\ldots.$ in the display of the calculator.
Thus we get the value of $\log_{4}24$ is $2.292$ to three decimals
The value of $\log_{4}24$ is $2.292$ .

Note: Using a calculator is a good and easy way to find the values of trigonometric functions , square values, cubic values, logarithmic values etc… . This change of base formula is especially helpful when using a calculator to evaluate a log to any base other than $10$ or $e$ because the calculator has options to calculate the logarithms with bases $10$ and $e$ only. The change of base formula can also be written in another, that is, $\log_{b}a\ \log_{c}b = \ \log_{c}a$ , which is also widely used in solving the problems.