# Solve the equation $22\log(92x+40.66)=38.9$

The equation $$22\log(92x+40.66)=38.9$$

steps so far

$$\log(92x+40.66)=\frac{38.9}{22}$$

to eliminate log, do I have to apply the opposite of log? Not sure what that is.

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Exponentiation is the oppisite of log – Amr Dec 5 '12 at 23:16
@Amr so I'm headed in the right direction, with doing the opposite? – Tyler Zika Dec 5 '12 at 23:17
Yes, exponentiate both sides. Where did the minus sign in front of $\log$ come from? – Ross Millikan Dec 5 '12 at 23:19

$$22\log(92x+40.66)=38.9\Longrightarrow \log(92x+40.66)=\frac{38.9}{22}\Longrightarrow$$

$$92x+40.66=e^{\frac{38.9}{2}}\Longrightarrow \,\,\ldots$$

If by $\,\log\,$ you mean logarithm in base $\,10\,$ just change $\,e\,$ for $\,10\,$

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In general, $$\log_{b}u=v \iff b^v=u.$$

Apply this fact. :-)

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You might have heard that $$b^{\log_b(x)} = x.$$ Most people that I am aware of just write $\log$ when they mean $\log_{10}$. So that would mean that $$10^{\log(x)} = x.$$ Now you have the equation $$\log(92x+40.66)=\frac{38.9}{22}$$ and to get rid of the $\log$ you can $$10^{\log(92x+40.66)}=10^{\frac{38.9}{22}}.$$ This is equivalent to $$92x + 40.66 = 10^{\frac{38.9}{22}}$$ and this is just a linear equation that you probably know how to solve (on both sides: subtract $40.66$ and then divide by $92$).

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