# Thread: Solving Equations using Logs

1. ## Solving Equations using Logs

Hi guys, I am unsure how this book has come about with the answer to this question as I thought I had it worked out.

Question:

Use the substitution y = 2^x to solve the following equation giving your answer in exact form involving logarithms to the base ten.

2^(2x) - 2^(x+3) + 15 = 0

My work goes like this:

y^2 - 2^(x+3) + 15 = 0
y^2 + 15 = 2^(x+3) <- Expand the log
log y^2 + log 15 = log 2^(x+3)
= (x+3)log 2
= xlog 2 + 3log2
= log 2^x + log 2^3
= log y + log 8 <- Rearrange the equation
log y^2 - log y = log 8 - log 15
log ((y^2)/y) = (log 8/15)
y = 8/15

The answers however state the following and I am unsure how they come about as I thought my working was sound.

log 3 / log 2 & log 5 / log 2

Any help would be greatly appreciated.

Cheers

Ben

2. You don't need logarithms.

$\displaystyle 2^{2x} - 2^{x + 3} + 15 = 0$

$\displaystyle \left(2^x\right)^2 - 2^3\cdot 2^x + 15 = 0$.

Now make the substitution suggested and solve the resulting quadratic.

3. Originally Posted by MuNch
Hi guys, I am unsure how this book has come about with the answer to this question as I thought I had it worked out.

Question:

Use the substitution y = 2^x to solve the following equation giving your answer in exact form involving logarithms to the base ten.

2^(2x) - 2^(x+3) + 15 = 0

My work goes like this:

y^2 - 2^(x+3) + 15 = 0
But you didn't make the substitution completely! $2^{x+ 3}= 2^x2^3= 8(2^x)$
Your equation should be $y^2+ 8y+ 15= 0$
which is a simple quadratic equation.

y^2 + 15 = 2^(x+3) <- Expand the log
log y^2 + log 15 = log 2^(x+3)
and this is also wrong: log(a+ b) is NOT equal to log(a)+ log(b).

= (x+3)log 2
= xlog 2 + 3log2
= log 2^x + log 2^3
= log y + log 8 <- Rearrange the equation
so NOW you have finished the substitution- but the left side, log(y^2)+ log(15), is still wrong.

log y^2 - log y = log 8 - log 15
log ((y^2)/y) = (log 8/15)
y = 8/15

The answers however state the following and I am unsure how they come about as I thought my working was sound.

log 3 / log 2 & log 5 / log 2

Any help would be greatly appreciated.

Cheers

Ben