Question

How do you solve \[{{3}^{2x}}=80\]?

Answer 1

Hint: Exponent is a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression like \[{{x}^{a}}\]. We use the formulae \[{{\left( {{a}^{m}} \right)}^{p}}={{a}^{mp}}\] for simplification.

Complete step by step answer:
As per the given question, we have to simplify the given expression. The given expression consists of exponents. So, we need to use the concept of exponents and the laws of exponents to simplify the given expression. Here, the given expression is
\[{{3}^{2x}}=80\]
One way of visualising basic positive integer exponent is as repeated multiplication
\[{{a}^{n}}=a\times a\times a\times a\left( n-times \right)\]
Then \[{{\left( {{a}^{b}} \right)}^{c}}\] means \[\left( {{a}^{b}} \right)\] multiplied c times. That is
 \[{{\left( {{a}^{b}} \right)}^{c}}=\left( {{a}^{b}} \right)\times \left( {{a}^{b}} \right)\times \left( {{a}^{b}} \right)\times ...c-times=\left( {{a}^{bc}} \right)\]
So, the given expression can be modified using the above-mentioned formulae.
 \[\Rightarrow {{3}^{2x}}={{\left( {{3}^{2}} \right)}^{x}}\]
We know that \[{{3}^{2}}\] can be written as \[3\times 3\] and its value is 9 now substituting this value in the above expression we get
\[\Rightarrow {{\left( {{3}^{2}} \right)}^{x}}={{\left( 9 \right)}^{x}}\]
On substituting the above value in the given equation. it becomes
\[\Rightarrow {{\left( 9 \right)}^{x}}=80\]
Now we apply ln on both sides. Then the equation becomes
\[\Rightarrow \ln {{\left( 9 \right)}^{x}}=\ln 80\]
According to the properties of logarithms \[\ln \left( {{a}^{x}} \right)=x\ln a\]. On applying the above formulae to the above equation
\[\Rightarrow x\ln 9=\ln 80\]
Now we divide with \[\ln 9\] on both sides. Then the equation becomes
\[\Rightarrow x\dfrac{\ln 9}{\ln 9}=\dfrac{\ln 80}{\ln 9}\]
\[\Rightarrow x=\dfrac{\ln 80}{\ln 9}\]
\[\Rightarrow x=1.99434627\]
Now we round off the value of x to 3 decimal places.
\[\Rightarrow x=1.994\]
Therefore, the value of x is \[1.994\].

Note:
In order to solve such a type of problem, we must have enough knowledge of the concept of exponents, logarithms, properties of logarithms, and laws of exponents. While solving the problems, we have to correctly express the exponents. We need to multiply the exponents or add up the powers wherever necessary. We should avoid calculation mistakes to get the desired results.