Question

How do you find the exact value of: ${\log _8}32 + {\log _8}2$ ?

Answer 1

Hint: In the above question, we are given a sum of two logarithmic functions with a specified base and we have to find the value of expressions. So to solve this question, we must know what logarithm functions are. A logarithm function is the inverse of an exponential function (a function in which one term is raised to the power of another term is known as an exponential function). An exponential function is of the form $a = {x^y}$ , so the logarithm function being the inverse of the exponential function is of the form $y = {\log _x}a$ . To evaluate this question, we will use the laws of the logarithm such as ${\log _a}b + {\log _a}c = {\log _a}bc$.

Complete step by step answer:
So, we will first condense the expression ${\log _8}32 + {\log _8}2$ using the laws of logarithms and then evaluate the final answer. So, we have, ${\log _8}32 + {\log _8}2$. We know the logarithmic property ${\log _a}b + {\log _a}c = {\log _a}bc$ that condenses the sum of two logarithmic functions and simplifies the expression. So, applying this logarithmic property in the expression, we get,
$\Rightarrow {\log _8}\left( {32 \times 2} \right)$
Carrying out the multiplication, we get,
$\Rightarrow {\log _8}64$

Now, we know that $64 = {8^2}$. So, we can write the value of $64$ as ${8^2}$ in the expression.
$\Rightarrow {\log _8}{8^2}$
Now, we also know another logarithmic property ${\log _a}{b^x} = x{\log _a}b$. So, applying this property into the expression, we get,
$\Rightarrow 2{\log _8}8$
Now, we know that ${8^1} = 8$. So, when we convert this exponential form into logarithmic form, we can say that ${\log _8}8 = 1$.So, we get,
$\therefore 2\left( 1 \right) = 2$

Hence, the exact value of ${\log _8}32 + {\log _8}2$ is $2$.

Note: The standard base of logarithm functions is ten, that is, if we are given a function without any base like $\log x$ then we take the base as 10. But in the given question, we are given the base as $8$. Now while applying the laws of the logarithm such as ${\log _a}b + {\log _a}c = {\log _a}bc$, we should keep in mind an important rule that is the base of the logarithm functions involved should be the same in all the calculations, as the base of both the functions in the question is the same, we can apply the logarithm laws in the given question.