Question

question 3: solve the exponential equation.
$10^{36} = 100^{18x}$
$x = 1$ option 1
$x = 36$ option 2
$x = \frac{1}{36}$ option 3
$x = \frac{4}{9}$ option 4

Explanation:

Step1: Rewrite 100 as a power of 10

Since \(100 = 10^2\), we can rewrite the right - hand side of the equation \(10^{36}=100^{18x}\) as \((10^{2})^{18x}\).
According to the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), \((10^{2})^{18x}=10^{2\times18x}=10^{36x}\). So the equation becomes \(10^{36}=10^{36x}\).

Step2: Solve for x

If two exponential functions with the same base \(a\) (\(a>0,a
eq1\)) are equal, i.e., \(a^{m}=a^{n}\), then \(m = n\).
In the equation \(10^{36}=10^{36x}\), the base \(a = 10\), \(m = 36\) and \(n = 36x\). So we have the equation \(36=36x\).
Divide both sides of the equation \(36 = 36x\) by 36: \(\frac{36}{36}=\frac{36x}{36}\), which simplifies to \(x = 1\).

Answer:

Option 1: \(x = 1\)