Question

Okay, I'm really fuzzy about my log and exponential chapter. For example, solving some exponential equation such as e^3x-7 x e^-2x=4e

I assume I start this equation by taking the natural log of both sides, but how will I know just by looking at an equation weather I need to find the log or the natural log?

Also, graphing. How to do you know where to put the base graph? Such as f(x)= log_1/2_x?

Answer 1

Usually you take logs to the base of the number having exponents. In your example, you have e^(3x-7), so take logs base e.

If you had written 2^(x+3) * 2^(x-7) = 32 then you'd take logs base 2.

As for graphing, just recall the all exponent/log graphs look the same, except for their slopes.

e^x looks just like 10^x. They both go through (0,1). e^x goes through (1,e) and 10^x goes through (1,10).

Fractional bases are just reflected through the y-axis, since (1/2)^x = 2^-x.

Now, logs look just like exponentials, but reflected through the origin, since if
y = e^x,
x = ln y

fractional bases are reflected through the x-axis.

So, as for log_1/2(x) It looks like log_2(x) but upside-down.

Visit wolframalpha.com and enter

log_4 x

or whatever base you want. It will also remind you that log_4(x) = lnx/ln4.

This also explains the reflection using fractional bases. log_1/2(x) = lnx/ln 1/2, and ln 1/2 is negative.