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Hello, I don't understand how to solve this function
f(x) = 6^x - 16 * 3^x

Hi, I will be happy to help.  I assume you are trying to find the x-intecepts, or when f(x) = 0.  If not, please let me know.

6^x - 16*3^x = 0

Since there are two different bases, you must use logarithms.
First, solve so each base is on it's own side of the equals sign.

6^x = 16 * 3^x

Next take the natural log, or ln, of both sides
ln6^x = ln(16*3^x)

Two rules of logarithms you need to use are

1) ln(a*b) = lna + lnb Or basically, if multiplying inside a logarithm, you can separate it into two separate logs by adding
2) ln(a^b) = b * lna Or basically bring out any exponents

ln6^x can be simplified, using 2), to xln6

ln(16*3^x) can first be simplified, using 1), to ln16 + ln(3^x)
Which then simplifies, using rule 2), to ln16 + xln3

So, the equation becomes

xln6 = ln16 + xln3

This can be solved like any linear equation by bringing all the variables, x, to one side.

xln6 - xln3 = ln16
x(ln6 - ln3) = ln16

Two more rules to use are
3) ln(a/b) = lna - lnb or basically, when dividing inside a log, the logs can be separated using subtraction
4) log(base a)b = lnb/lna or the change of base formula

The left side of the equation can be simplified using 3) to get
xln(6/3) = ln16
xln2 = ln16

Dividing both sides by ln2 gets
x = ln16/ln2

Which simplifies, using the change in base formula,
x = log(base2)16


x = 4


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Anne Losch


Solving equations, graphing, evaluatin equations, factoring, functions, systems of equations, rational equations, exponent, complex numbers, word problems, logarithms, polynomials, and all topics in an Algebra 1, Algebra 2, or College Algebra class


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