Question

How do you solve \[{{3}^{x}}=500\]?

Answer 1

Hint: For solving such questions we have to take log on both sides of the equation and then use logarithmic power property to eliminate the power of exponent then solve the equation to get the final answer.
We have to apply the following formula to solve the question.
${{\log }_{e}}\left( {{p}^{q}} \right)=q{{\log }_{e}}\left( p \right)$

Complete step by step solution:
The given equation is
${{3}^{x}}=500$
For solving such type of questions we have to take ${{\log }_{e}}$on both side we get
${{\log }_{e}}\left( {{3}^{x}} \right)={{\log }_{e}}\left( 500 \right)$
Now we have to use logarithmic power property to eliminate the power of the exponent in the above equation. Therefore, the above expression gets the following form
$x{{\log }_{e}}\left( 3 \right)={{\log }_{e}}\left( 500 \right)$
Now divide the above equation by ${{\log }_{e}}\left( 3 \right)$ we get
$x=\dfrac{{{\log }_{e}}\left( 500 \right)}{{{\log }_{e}}\left( 3 \right)}$
Now from the log table we have to calculate the values of the above logarithmic terms. Therefore,
${{\log }_{e}}\left( 500 \right)=6.2146$
And
${{\log }_{e}}\left( 3 \right)=1.0986$
Therefore, the final answer is
$x=\dfrac{6.2146}{1.0986}$
After solving the above expression we get
$x=5.6568$
Hence, the final answer of the given problem is
$x=5.6568$

Additional information:
In such types of questions it is very important to apply the correct property at the correct place. Suppose we have to apply product property then we have to apply the product property of the logarithmic.
Similarly, if we require a product then in that case we have to apply the product rule of the logarithmic.
The following are the main properties of the logarithmic:
Product rule:
${{\log }_{e}}\left( xy \right)={{\log }_{e}}\left( x \right)+{{\log }_{e}}\left( y \right)$
Quotient rule:
${{\log }_{e}}\left( \dfrac{x}{y} \right)={{\log }_{e}}\left( x \right)-{{\log }_{e}}\left( y \right)$
Power rule:
${{\log }_{e}}\left( {{p}^{q}} \right)=q{{\log }_{e}}\left( p \right)$

Note:
In this question we have seen that we just need to memorize the properties of logarithmic functions and then apply them correctly to solve these questions. It is also important for a student to do the calculation carefully.