Question

How do you use the change of base formula and a calculator to evaluate the logarithm ${\log _8}77?$

Answer 1

Hint : As we know that the logarithm is the inverse function to exponentiation. That means the logarithm of a given number $x$ is the exponent to which another fixed number, the base b, must be raised , to produce that number $x$ . As per the definition of a logarithm ${\log _a}b = c$ which gives that ${a^c} = b$ . Here in the above expression the base is $8$ . And we also have to assume that if no base $b$ is written then the base is always 10. This is an example of base ten logarithm because $10$ is the number that is raised to a power.

Complete step-by-step answer :
As per the given question we have ${\log _8}77.$ .
We can use a formula that allows us to rewrite a logarithm in terms of logs written with another base. The change of base formula is given as
${\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}$ , in which value of $a = 8$ and $b = 77$ .
So by applying the formula we have ${\log _8}77 = \dfrac{{\log 77}}{{\log 8}}$ .
It can be written as $\dfrac{{\ln 77}}{{\ln 8}}$ , so it gives us ${\log _8}77 = \dfrac{{4.34381}}{{2.079442}} = 2.08893$
Hence the value of ${\log _8}77$ is $2.0889$ (approx.).

Note : We should always be careful while solving logarithm formulas and before solving this kind of problems we should know all the rules of logarithm and exponentiation. We have to keep in mind that when a logarithm is written without any base, like this: $\log 100$ then this usually means that the base is already there which is $10$ . It is called a common logarithm or decadic logarithm, is the logarithm to the base $10$ . One way we can approach log problems is to keep in mind that ${a^b} = c$ and ${\log _a}c = b$ .