Simplying Log Problems

**metallica007** · January 11th 2010, 11:20 PM

Hi My Friends
I need ur help to simply the following problem

can u please explain the steps to solve the problem

Thanks in Advanced

**Grandad** · January 12th 2010, 12:15 AM

Hello metallica007

Originally Posted by metallica007

Hi My Friends
I need ur help to simply the following problem

can u please explain the steps to solve the problem

Thanks in Advanced

The definition of a log is this (and it's a bit involved!):

The log of a number to a certain base is that power to which the base must be raised in order to get the number.

In other words, if

$x = \log_b(y)$

then the log of $y$ to base $b$ is $x$ . So when we raise $b$ to the power $x$ we get $y$ . In other words:

$b^x = y$

Make sure you can get your mind around that!

The laws of logs (whatever the base) are:

$\log (a) + \log (b) = \log(ab)$ ... (1)

$\log(a) -\log(b) = \log\left(\frac{a}{b}\right)$ ... (2)

$a\log(b) = \log\left(b^a\right)$ ... (3)

Now to your question. Note first that $\sqrt5 = 5^{\frac12}$ . So we get:

$\log_5\left(\frac{\sqrt5}{5}\right) = \log_5(\sqrt5) - \log_5(5)$ , using law (2) above

$=\log_5(5^{\frac12}) - \log_5(5^1)$

$=\tfrac12-1$ , using the definition of a log

$=-\tfrac12$

Grandad

**metallica007** · January 12th 2010, 08:12 AM

Originally Posted by Grandad

Hello metallica007The definition of a log is this (and it's a bit involved!):