Question

How does one solve $8({10^{3x}}) = 12$ ?

Answer 1

Hint:For solving the particular problem we firstly divide our expression by $8$ and simplify it then take the base $10$ logarithm function of both sides of the equation to remove the variable $x$ from the exponent or power. After that simplify the further equation and after simplification we can get the corresponding result.

Formula Used:
In this particular question we used following logarithmic functions
${\log _a}{b^x} = x{\log _a}b$
Where $a$is the base ,
And $x$ is the exponent of $b$.
We also used another property of logarithm i.e.,
$\log \dfrac{a}{b} = \log a - \log b$
We also used another property of logarithm i.e.,
${\log _a}a = 1$
Log of something with the same base is always unity.

Complete step by step answer:
We have $8({10^{3x}}) = 12$ ,
Divide the expression by $8$ and simplify!
$\Rightarrow 8({10^{3x}}) = 12 \\
\Rightarrow {10^{3x}} = \dfrac{{12}}{8} \\
\Rightarrow {10^{3x}} = \dfrac{3}{2} \\ $
for solving $x$ which is in the power then, we use the rule of log function,
therefore, taking log with a base $10$ both the side ,
$\Rightarrow {\log _{10}}{10^{3x}} = {\log _{10}}\dfrac{3}{2} \\
\Rightarrow 3x{\log _{10}}10 = {\log _{10}}3 - {\log _{10}}2 \\
\Rightarrow 3x = 0.4771 - 0.3010 \\
\Rightarrow x = \dfrac{{0.1761}}{3} \\
\Rightarrow x = 0.0587$
Here we get our answer by simplifying and using certain logarithm functions .

Additional Information:
We can simply say that the logarithm counts the number of occurrences of the same factor in repeated multiplication for instance take \[1000{\text{ }} = {\text{ }}10\, \times \,10\, \times \,10{\text{ }} = {\text{ }}{10^3}\]. The "logarithm with base $10$ " of $1000$ is $3$.

Note:The logarithm is also known as inverse function to exponentiation. When we apply the logarithm and the variable $x$ is the power or exponent to which another fixed number or constant, the base $b$, must be raised, to produce that number $x$.
The logarithm of $x$ with base $b$is represented by ${\log _b}(x)$ ,or not using parentheses, ${\log _b}x$ , or we can also write like, $\log x$ .