   SEARCH HOME Math Central Quandaries & Queries  Todd, a student: 7(6^x)=6e^-2x Hi Todd,

Let me show you a similar question: 3(2x+4)=5ex

Since there is no way we can change the bases of the exponential we need to take the logarithm of both sides of the equations (For a review: the relationship between logarithms and exponential functions.) I will use the natural logarithm where ln is the same thing as loge because there is an e in my equation.

ln[3(2x+4)]=ln(5ex)

Using the law of logarithms where log ab=log a + log b, I simplify the equation:

ln 3 + ln (2x+4) = ln 5 + ln(ex)

Using the law of logarithms where log ab = b log a, I will simplify further:

ln 3 + (x+4)(ln 2) = ln 5 + x (ln e)

I will distribute ln 2 through x+4 and simplify ln e since it equals 1:

ln 3 + x ln2 + 4 ln2 = ln 5 + x

Now I will collect the terms with variables to one side of the equation and those without to the other side:

x ln2 - x = ln 5 -ln 3 - 4 ln2

I need to get x by itself so I will factor it out:

x (ln2 - 1) = ln 5 -ln 3 - 4 ln2

Now I will divide both sides by ln2 -1:

x = (ln 5 -ln 3 - 4 ln2)/(ln2 -1)

You can keep this value of x for an exact value or enter it into your calculator to find the approximate value.

Hope this helps,

Janice     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.