Question

How do you solve ${\log _2}\left( {24} \right) - {\log _2}\left( 3 \right) = {\log _5}\left( x \right)$

Answer 1

Hint : In the question, we are provided with a logarithm expression. Apply the logarithm formulas carefully which are listed below and solve the question and hence we got our required answer. The formulas to be used are ${\log _a}b - {\log _a}c = {\log _a}\left( {\dfrac{b}{c}} \right)$ and then $\log \left( {{a^b}} \right) = b\log \left( a \right)$ and then finally ${\log _a}\left( a \right) = 1$

Complete step-by-step answer :
For solving such types of questions, you must have a good command on the logarithm properties.
The given expression is ${\log _2}\left( {24} \right) - {\log _2}\left( 3 \right) = {\log _5}\left( x \right)$ and we have to find the value of “x”.
Firstly, using the property,
${\log _a}b - {\log _a}c = {\log _a}\left( {\dfrac{b}{c}} \right)$ which states that if the base of two logarithm quantities (“b” and “c”) are same which is “a” and they had a subtraction sign in between then the solution would be logarithm of the same base with the division of these two quantities.
Applying this formula in the given expression
$\Rightarrow {\log _2}\left( {\dfrac{{24}}{3}} \right) = {\log _5}\left( x \right) \\\ \Rightarrow {\log _2}\left( 8 \right) = {\log _5}\left( x \right) \;$
Now, $8 = {2^3}$ replacing the first term by this quantity.
${\log _2}\left( {{2^3}} \right) = {\log _5}\left( x \right)$
And also using the property of logarithm that
$\log \left( {{a^b}} \right) = b\log \left( a \right)$ which states that if we want the logarithm of the power value then the power would be multiplied first to the logarithm of the base.
$3{\log _2}\left( 2 \right) = {\log _5}\left( x \right)$
And one more property which states that when the base and the quantity of the logarithm are same then the value is one. ${\log _a}\left( a \right) = 1$
$3\left( 1 \right) = {\log _5}\left( x \right)$
Taking antilog
$\Rightarrow x = {5^3} = 125$
So, the required answer is $125$
So, the correct answer is “125”.

Note : All the logarithm properties should be learnt on tips. You don’t need to open the formula sheet for solving such basic questions. The properties are very easy to memorize, you just need to practice a lot. learn the formulas carefully. Remember log functions are inverse of exponential functions.