Question

How do you solve \[{{e}^{3x}}=5\]?

Answer 1

Hint: In this problem, we have to solve the given exponential expression and find the value of the unknown variable x. We know that to solve these types of problems we have to know some exponential and logarithmic formulas and properties. We can first multiply log on both the sides and then solve by using logarithmic properties to find the value of x. In this problem we are going to use two properties of logarithm.

Complete step by step answer:
We know that the given exponential expression to be solved is,
\[{{e}^{3x}}=5\]
We can now multiply log on both sides, we get
\[\Rightarrow \log {{e}^{3x}}=\log 5\]
We know that the logarithmic property is,
\[\log {{\left( x \right)}^{a}}=a\log \left( x \right)\]
We can apply the above logarithmic properties in the above step, we get
\[\Rightarrow 3x\log \left( e \right)=\log 5\]
We also write the below logarithmic property in the above step,
\[{{\log }_{a}}a=1\]
We can apply the above property in the above step, we get
\[\Rightarrow 3x\times 1=\log 5\]
Now we can divide by 3 on both sides, we get
\[\Rightarrow \dfrac{3x\times 1}{3}=\dfrac{\log 5}{3}\]
We can now cancel the similar terms to get,
\[\Rightarrow x=\dfrac{\log 5}{3}\]
Therefore, the value of x is \[\dfrac{\log 5}{3}\].

Note:
Students make mistakes while writing the properties of logarithm. We should know some logarithmic and exponential conversions, properties, formulas to solve these types of problems. In this problem, we have used several logarithmic properties which should be remembered to solve these types of problems. We can also find the exact value of the unknown variable by using scientific calculators.