Question

How do you calculate ${{\log }_{8}}512?

Answer 1

Hint: The logarithmic function is an inverse of the exponential function. It is defined $y={{\log }_{a}}x$ if and only if $x={{a}^{y}}$ for $x>0$ $a>0$ and $a\ne 1$ we can solve by exponential logarithm format. $y=\log bxy$ is an exponent $b$ is called the base and $x$ is the number that results from raising the $b$ to the power of $y$ an equivalent is ${{b}^{y}}=x$.

Complete step-by-step answer:
We have ${{\log }_{8}}512$ as per question exponential logarithm format $y={{\log }_{b}}x$
So, $y={{\log }_{8}}512$ Here base is $8$ and $x$ is the number that result from raising the base to the power of $y.$
${{b}^{y}}=x$
${{8}^{y}}=512$ this is in exponential form.
${{8}^{y}}={{8}^{3}}$
$y=3$
$8\times 8\times 8=512$

${{\log }_{8}}512=3$
So, we can calculate this just by using logarithms law.

Additional Information:
We can solve this example with another method as given below.
${{\log }_{8}}512=\dfrac{\log 512}{\log 8}=\dfrac{\log {{2}^{9}}}{\log {{2}^{3}}}=\dfrac{9.\log 2}{3.\log 2}$
$=\dfrac{9.\log 2}{3.\log 2}=\dfrac{9}{3}=3$
So, we get the same answer i.e. ${{\log }_{8}}=512=3$
This method or solution has the same answer. As compared to the before method. This solution is based on the different logarithm laws.
The $\log $ can be calculated by using a calculator or by using a log table.
Much power of logarithms is useful in solving exponential equations. Some examples of this include sound (decimal measures) earthquakes (Richter Scale) the brightness of star and chemistry (pH balance, a measure) of acidity and alkalinity.

Note:
Mathematicians use the notation $\ln \left( x \right)$ to indicate the natural logarithm of a positive number. Most have buttons for $\ln $ and log which denotes logarithm base $10$ so you can compute logarithms in base or base $10$. So, while solving log problems this.
The natural logarithm of number is its logarithm to the base of the mathematical constant where $e$ is an irrational transcendental number approximately equal to $2.718281828.$ for converting $\ln $ to $\log $ use the equation $\ln \left( x \right)=\log x\div \log \left( 2.71828 \right)$ use different log rules while solving the numerical on logarithm try not to make mistakes in formula because so many students make mistakes on formula.