 # Find the solution log(6x+10) = log(x)/log(1/2) Zoe Oneal 2021-01-31 Answered
Find the solution $\mathrm{log}\left(6x+10\right)=\frac{\mathrm{log}\left(x\right)}{\mathrm{log}\left(\frac{1}{2}\right)}$
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The solution is $x=0.464343$ approx. using the Intermediate Value Theorem.
METHOD
The quantity on the right can only be positive when 0 The domain can be split into 0.1 intervals, and by substituting $x=0.1$ to 0.9 in the expression $\mathrm{log}\left(6x+10\right)-\frac{\mathrm{log}\left(x\right)}{\mathrm{log}\left(\frac{1}{2}\right)}$ where log is assumed to be base 10, it can be seen that the expression changes sign between 0.4 and 0.5.
This interval can be split into hundredths: and re-evaluated. By continuing this process an approximate value of x can be found.
If the logs are natural logs $x=0.188211$ approx.
Note that $\frac{1}{{\mathrm{log}}_{10}\left(\frac{1}{2}\right)}=\mathrm{log}\left[base\frac{1}{2}\right]\left(10\right)$, so the question can be expressed: