Question

Evaluate the value of ‘x’ such that \[{5^{x - 3}} \times {3^{2x - 8}} = 225\].

Answer 1

Hint: In this question, we need to evaluate the value of ‘x’ such that \[{5^{x - 3}} \times {3^{2x - 8}} = 225\]. For this, we will use the properties of the exponential function along with the general algebraic operations.

Complete step-by-step answer:
Given the number \[{5^{x - 3}} \times {3^{2x - 8}} = 225\]
Now we will first factorize the RHS of the given number in same form as LHS as
\[
   5\underline {\left| {225} \right.} \\
   5\underline {\left| {45} \right.} \\
   3\underline {\left| 9 \right.} \\
   3 \;
 \]
Hence we can write
\[
\Rightarrow \left( {225} \right) = 5 \times 5 \times 3 \times 3 \\
   = {5^2} \times {3^2} \;
 \]
By substituting the values we can rewrite the number as
\[{5^{x - 3}} \times {3^{2x - 8}} = {5^2} \times {3^2}\]
Now we will compare the both sides of the number and as we know if the bases are the same then the exponents must be equal so from the number we can see both side have the same base as 3 and 5 on the both sides hence we can write
\[
\Rightarrow {5^{x - 3}} = {5^2} \\
\Rightarrow x - 3 = 2 \\
\Rightarrow x = 5 \;
 \]
Also we can write
\[
\Rightarrow {3^{2x - 8}} = {3^2} \\
\Rightarrow 2x - 8 = 2 \\
\Rightarrow 2x = 10 \\
\Rightarrow x = 5 \;
 \]
Hence the value of \[x = 5\]
So, the correct answer is “x=5”.

Note: An exponential equation is one in which a variable occurs in the exponent, in an exponential equation each side can be expressed in terms of the same base and can be solve by using the property
\[
  {a^x} = {a^y} \\
  x = y \;
 \]