Question

How do you determine the base when solving for x by converting to a log?

Answer 1

$x=\frac{\ln \left(\frac{3}{2}\right)}{4 \ln (1.0125)}$

Solution:

Given expression:

$(1.0125)^{4 x}=\frac{3}{2}$

To solve the expression:

$(1.0125)^{4 x}=\frac{3}{2}$

If f(x) = g(x) then ln(f(x)) = ln(g(x)).

Using the above condition, we can write

$\ln \left(1.0125^{4 x}\right)=\ln \left(\frac{3}{2}\right)$

Apply log rule: $\log _{a}\left(x^{b}\right)=b \cdot \log _{a}(x)$

$4 x \ln (1.0125)=\ln \left(\frac{3}{2}\right)$

Divide both side of the equation by $4 \ln (1.0125)$.

$\frac{4 x \ln (1.0125)}{4 \ln (1.0125)}=\frac{\ln \left(\frac{3}{2}\right)}{4 \ln (1.0125)}$

$x=\frac{\ln \left(\frac{3}{2}\right)}{4 \ln (1.0125)}$

The answer is $x=\frac{\ln \left(\frac{3}{2}\right)}{4 \ln (1.0125)}$.