1. ## Logorithms

Hi! I'm afraid I really don't understand logorithms. If someone showed me how to answer the following questions I'd really help. Thanks!

logy = 3x +2

Find x when y = 500

Find y when x = -1

Express log(y^4) in terms of x

Find an expression for y in terms of x

And the logs are to base 10.

2. Hello Tesphen
Originally Posted by Tesphen
Hi! I'm afraid I really don't understand logorithms. If someone showed me how to answer the following questions I'd really help. Thanks!

logy = 3x +2

Find x when y = 500
Just put $\displaystyle 500$ into your calculator, press the log button, and solve for $\displaystyle x$:

$\displaystyle 2.6990 = 3x + 2$

$\displaystyle \Rightarrow$$\displaystyle x = 0.2330$

Find y when x = -1
When $\displaystyle x=1$, $\displaystyle \log y = -3 + 2 = -1$

Now you need to use the definition of a logarithm, which is:

• The log (to any base) of a number is that power to which the base must be raised in order to obtain the number.

I know, confusing isn't it? So let's unpack it here. We have

$\displaystyle \log_{10} y = -1$

So $\displaystyle -1$ is the power to which the base $\displaystyle (10)$ must be raised to obtain the number (that's $\displaystyle y$). So we have:

$\displaystyle y = 10^{-1} = 0.1$

It's really simple when you know how!

Express log(y^4) in terms of x
Here we use the law of logs: $\displaystyle \log(a^b)= b\log(a)$. So:

$\displaystyle \log(y^4) = 4\log(y) = 4(3x+2)$

And that's it!

Find an expression for y in terms of x
Again, we use the definition of a log that I've given you above. So here we have

$\displaystyle \log_{10}(y) = 3x + 2$

In other words, the log to base $\displaystyle 10$ of $\displaystyle y$ is $\displaystyle (3x+2)$. So, using the definition, $\displaystyle (3x+2)$ is the power to which we must raise $\displaystyle 10$ to get $\displaystyle y$. So

$\displaystyle y = 10^{(3x+2)}$

And that's your expression for $\displaystyle y$ in terms of $\displaystyle x$. I hope that helps to make it clearer!