dream13rxs

2022-06-29

How do you solve logarithmic equations like this one?
How do you solve
$3\mathrm{log}\left(x-15\right)={\left(\frac{1}{4}\right)}^{x}?$
The solution is approximately 16. How would you solve a logarithmic equation with an solution approximately equal to a number without using a graphing calculator?

### Answer & Explanation

engaliar0l

You don't say what base of logs you are using. The approach will be the same in any case-I will assume natural logs. We must have $x>15$ or the logarithm is not defined. In that case, the right side will be very small and positive. We need $x>16$ to make the left side positive. Define $a=x-16$, where we expect $a$ to be very small, so we will use the first term of the Taylor series.
$3\mathrm{log}\left(x-15\right)={\left(\frac{1}{4}\right)}^{\phantom{\rule{negativethinmathspace}{0ex}}x}\phantom{\rule{0ex}{0ex}}3\mathrm{log}\left(1+a\right)={\left(\frac{1}{4}\right)}^{\phantom{\rule{negativethinmathspace}{0ex}}16+a}\phantom{\rule{0ex}{0ex}}3\cdot {4}^{16}a={4}^{-a}$
This shows $a\approx \frac{1}{3\cdot {4}^{16}}$ with both sides very close to $1$, so $x\approx 16+\frac{1}{3\cdot {4}^{16}}$

Jameson Lucero

The point is that for $x>10$ the right hand side is smaller than $1/1000000,$ so you are essentially solving for $LHS=0.$ There is no general method if you are not so lucky.

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