Question

Given log2 = 0.3010300, log3 = 0.4771213, log7= 0.8450980, find the value of log64.

Answer 1

Hint: In this question we will use the property of exponent in logarithm to evaluate the value. First, we will convert the number inside the log i.e. 64 into the form of the exponent. After this, we will use the property that $\log {x^a}= alogx$, where x is a natural number. Finally, we will put the value of log2 to get the answer.

Complete step-by-step solution:
The given question is log64 and we have to calculate the numerical value of the logarithm.
The first thing that we do is to convert the given expression in the form where we can put the given values.
The number 64 can be written in exponent form as follow:
$64 = {2^6}$, Where ‘2’ is the base and ‘6’ is the exponent.
Now, we know the property related to power in logarithm which is given as:
$\log {x^a}= alogx$, where ‘x’ is a natural number.
Using this property we have:
$log64 = log{2^6} = 6log2$.
It is given that log2 = 0.3010300.
Putting this value in above expression, we get:
$6log2 = 6 \times 0.3010300 = 1.80618.$
So we can say that log 64 = 1.80618.

Note: In this type of question, you should know to convert the number in exponent form. First, find the prime factor of the given number then write in product form and finally in exponent form. For example- 125 = $5 \times 5 \times 5 = {5^3}$. You should also remember other properties related to the log.
1.$log(x.y) = logx + logy$
2. $log(x/y) = logx – logy$