Question

Solve the following equation, having given log2 =0.30 and log 5 =0.7.
${5^{5 - 3x}} = {2^{x + 2}}$ .

Answer 1

Hint: In this question, we proceed by taking logs on both sides. After taking a log, the expression gets simplified. So, we will then rearrange the terms and then put the value of log5 and log2 to finally get the value of x.

Complete step-by-step answer:
The given expression is
${5^{5 - 3x}} = {2^{x + 2}}$.
This is the equation as there LHS and RHS are separated by the equality sign.
We will first take logs on both sides.
So, taking log on both sides, we get:
$\log {5^{5 - 3x}} = \log {2^{x + 2}}$
$ \Rightarrow (5-3x) \log{5} = (x +2) \log{2}$.
Putting the values of log5 and log2 in above equation, we get:
$(5 -3x)0.7 = (x+2)0.3$
On solving this equation, we have:
$ \Rightarrow 3.5 – 2.1x = 0.3x +0.6$
On rearranging the terms, we get:
$ \Rightarrow 0.3x + 2.1x = 3.5 – 0.6 = 2.9$
$ \Rightarrow 2.4x = 2.9$
On dividing by 2.4 on both sides, we get:
$ \Rightarrow x = \dfrac{{2.9}}{{2.4}} = 1.208$ .
Therefore, the solution of the given equation is $x = 1.208$.

Note: In the question related to logarithm , you should remember the properties related to different mathematical operations on log. Some of the important properties are as follow:
log(xy) = logx + logy.
log(x/y) = logx – logy.
$\log {x^y}$ = y logx.
You should know that log is not defined for zero. The value of log1 = 0. Logarithm is not defined for negative numbers. In the problem related to exponent where we have a different base on two sides, taking log on both sides becomes a crucial step which simplifies the equation.