Question

If we have $\log 5 = a$, what is $\log 500$?

Answer 1

To find the value of $\log 500$, we first write $500$ as $\left( {5 \times 100} \right)$. Then we will use the product rule to simplify it. We will write $100$ as ${10^2}$ and then we will use the power rule. At last, using the given data i.e., $\log 5 = a$ we will further simplify it to find the result.

Complete step-by-step solution:
We have to find the value of $\log 500$.
We can write $\log 500$ as $\log \left( {5 \times 100} \right)$ i.e.,
$\Rightarrow \log \left( {500} \right) = \log \left( {5 \times 100} \right)$
Now using the product rule i.e., ${\log _b}(mn) = {\log _b}(m) + {\log _b}(n)$, we can write
$\Rightarrow \log \left( {500} \right) = \log \left( 5 \right) + \log \left( {100} \right)$
We can write $\log \left( {100} \right)$ as $\log \left( {{{10}^2}} \right)$. So, we get
$\Rightarrow \log \left( {500} \right) = \log \left( 5 \right) + \log \left( {{{10}^2}} \right)$
Using the Power rule i.e., ${\log _b}\left( {{m^p}} \right) = p{\log _b}m$ we can write
$\Rightarrow \log \left( {500} \right) = \log \left( 5 \right) + 2\log \left( {10} \right)$
Given, $\log 5 = a$ and also, we know that ${\log _a}a = 1$
Using this we get
$\Rightarrow \log \left( {500} \right) = a + 2$
Therefore, if $\log 5 = a$ then $\log 500$ is $\left( {a + 2} \right)$.
Additional information:
In logarithm to get a certain number, the power is raised to some number which is typically a base number. Logarithmic functions are inverse of exponential functions and we can express any exponential function in logarithmic form. Similarly, all logarithmic functions can be rewritten in exponential form.
Properties of Logarithmic functions are as follows:
$(1)$ The Product Rule: ${\log _b}(mn) = {\log _b}(m) + {\log _b}(n)$
This property denotes that the $\log$ of a quotient is the difference of the $\log$ of the dividend and the divisor.
$(2)$ The Quotient Rule: ${\log _b}\dfrac{m}{n} = {\log _b}m - {\log _b}n$
This property denotes that the $\log$ of a quotient is the difference of the $\log$ of the dividend and the divisor.
$(3)$ The Power Rule: ${\log _b}\left( {{m^p}} \right) = p{\log _b}m$
This property denotes that $\log$ of a power is the exponent times the logarithmic of the base of the power.
$(4)$ The Zero Exponent Rule: ${\log _b}1 = 0$
This property denotes that $\log$ of $1$ is zero.
$(5)$ Change of Base Rule: ${\log _b}x = \dfrac{{{{\log }_a}x}}{{{{\log }_a}b}}$
This property denotes that we can rewrite the logarithm as the quotient of logarithm of any other base.

Note: It is very important to note that $\log x$ denotes that the base is $10$ and $\ln x$ denotes that base is $e$. Also, ${\log _b}x$ is only defined when $b$ and $x$ are two positive real numbers and b is not equal to $1$. Logarithm base $10$ i.e., $b = 10$ is called decimal or common logarithm, logarithm base $e$ is called natural logarithm and binary logarithm uses base $2$.