Question

How do you solve ${3^{2x}} = 75$?

Answer 1

Hint: This question is in the form of exponents, we will take the log on both the sides and further simplify it to get the value of $x$.

Complete step-by-step solution:
We have the given equation as: ${3^{2x}} = 75$
On taking the logarithm on both the sides, we get:
$ \Rightarrow {\log _{10}}{3^{2x}} = {\log _{10}}75$
Now we know the property of logarithm that${\log _{10}}{a^b} = b{\log _{10}}a$, we can write the equation as:
$ \Rightarrow (2x) \times {\log _{10}}3 = {\log _{10}}75$
Now on dividing both the sides of the expression with ${\log _{10}}3$, we get:
$ \Rightarrow \dfrac{{(2x) \times {{\log }_{10}}3}}{{{{\log }_{10}}3}} = \dfrac{{{{\log }_{10}}75}}{{{{\log }_{10}}3}}$
On simplifying, we get:
$ \Rightarrow 2x = \dfrac{{{{\log }_{10}}75}}{{{{\log }_{10}}3}}$
On dividing the expression with $2$ on both the sides, we get:
$ \Rightarrow \dfrac{{2x}}{2} = \dfrac{1}{2} \times \dfrac{{{{\log }_{10}}75}}{{{{\log }_{10}}3}}$
On simplifying, we get:
$ \Rightarrow x = \dfrac{1}{2} \times \dfrac{{{{\log }_{10}}75}}{{{{\log }_{10}}3}}$
Now on using the scientific calculator to get the values, we get:
$ \Rightarrow x = \dfrac{1}{2} \times \dfrac{{1.875061}}{{0.47712}}$
On simplifying, we get:
$ \Rightarrow x = 1.96497$

Therefore the value of x is 1.96497.

Note: Now to cross check whether the answer of $x$ is correct, we will put the value of $x$ in the left-hand side of the expression to get the value, on substituting, we get:
$ \Rightarrow {3^{2(1.96497)}}$
On multiplying the exponent term, we get:
$ \Rightarrow {3^{3.92994}}$
On using the scientific calculator, we get:
$ \Rightarrow 75$, which is the right-hand side, therefore the solution is correct.
It is to be noted that the logarithm we are using has the base$10$, the base is the number to which the log value has to be raised to, to get the original term. This is also called the antilog of the number which is the logical reverse of taking a log.
The most commonly used bases in logarithm are $10$ and $e$ which has a value of approximate $2.713...$
Logarithm is used to simplify a mathematical expression; it converts multiplication to addition, division to subtraction and exponents to multiplication.