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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

Or

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

Or

xln2 = ln16

Dividing both sides by ln2 gets

x = ln16/ln2

Which simplifies, using the change in base formula,

x = log(base2)16

Or

x = 4

Algebra

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