Question

How do you solve $5\cdot {{18}^{6x}}=26$?

Answer 1

Hint: For solving the given exponential equation, we first have to divide both sides of the equation by $5$ so that we can separate the variable term on one side. Then we have to consider the logarithms on both the sides of the equation obtained so that we can take out the variable from the exponent over $18$. For this we need to use the logarithm property given by $\log {{a}^{m}}=m\log a$. Finally, using simple algebraic operations we can obtain the required solution of the given equation.

Complete step by step solution:
The equation given in the above question is written as
$\Rightarrow 5\cdot {{18}^{6x}}=26$
On dividing the above equation by $5$, we get
$\begin{align}
  & \Rightarrow \dfrac{5\cdot {{18}^{6x}}}{5}=\dfrac{26}{5} \\
 & \Rightarrow {{18}^{6x}}=\dfrac{26}{5} \\
\end{align}$
On simplifying the RHS, we get
$\Rightarrow {{18}^{6x}}=5.2$
Now, taking the logarithm of base $18$ on both the sides, we get
$\Rightarrow {{\log }_{18}}{{18}^{6x}}={{\log }_{18}}5.2$
Now, using the logarithm property given by $\log {{a}^{m}}=m\log a$ we can write the above equation as
$\Rightarrow 6x{{\log }_{18}}18={{\log }_{18}}5.2$
Now, we know that ${{\log }_{a}}a=1$. So the above equation becomes
\[\begin{align}
  & \Rightarrow 6x\left( 1 \right)={{\log }_{18}}5.2 \\
 & \Rightarrow 6x={{\log }_{18}}5.2 \\
\end{align}\]
Finally, on dividing the above equation by \[6\], we get
$\Rightarrow x=\dfrac{{{\log }_{18}}5.2}{6}$

Hence, we have finally obtained the solution of the given equation as $x=\dfrac{{{\log }_{18}}5.2}{6}$.

Note: We need to remember the important properties of the logarithm function in order to solve these types of questions. We may further obtain the obtained solution in the decimal form. But for that we need a scientific calculator since the logarithm term is not easy to evaluate.