# Log Laws [Solved!]

### My question

I refer to 3. Logarithm Laws

For this email I will use a convention where if a^x = b then x is written as log(a) b. i.e. the base of the log is written in parantheses. So if for example we have (ab)^c = d then c is given by log(ab) d.

Now I think that log(xy) z can be rewritten as following

log(xy) z = 1/[(1/log(x) z) + (1/log(y) z)]

Have you heard of such an identity?

### Relevant page

3. Logarithm Laws

### What I've done so far

X

I refer to <a href="/exponential-logarithmic-functions/3-logarithm-laws.php">3. Logarithm Laws</a>

For this email I will use a convention where if a^x = b then x is written as log(a) b. i.e. the base of the log is written in parantheses. So if for example we have (ab)^c = d then c is given by log(ab) d.

Now I think that log(xy) z can be rewritten as following

log(xy) z = 1/[(1/log(x) z) + (1/log(y) z)]

Have you heard of such an identity?
Relevant page

<a href="/exponential-logarithmic-functions/3-logarithm-laws.php">3. Logarithm Laws</a>

What I've done so far

I've been thinking about this for some time now.

## Re: Log Laws

Hi Michael

It is quite hard to read your question. You are encouraged to use the math entry system.

Actually, I have not seen this before, but it is true. I am using change of base formula (which is on this page: 5. Natural Logarithms (base e)

I am changing to base 10, so I just write "log" (but I could change it to any base).

\text{LHS}

= \log_{xy} z

= \frac {log z}{\log xy}

= \frac {log z}{\log x + \log y}

\text{RHS}

= \frac{1}{(\log x / \log z)} + \frac{\log y}{\log z}

= \frac{1}{(\log x + \log y)/\log z}

= \frac{\log z}{\log x + \log y}

Phew!

I'm not sure what you would use it for, though!

Regards

X

Hi Michael

It is quite hard to read your question. You are encouraged to use the math entry system.

Actually, I have not seen this before, but it is true. I am using change of base formula (which is on this page: <a href="/exponential-logarithmic-functions/5-logs-base-e-ln.php">5. Natural Logarithms <span class="noWrap">(base e)</span></a>

I am changing to base 10, so I just write "log" (but I could change it to any base).

\text{LHS}

= \log_{xy} z

= \frac {log z}{\log xy}

= \frac {log z}{\log x + \log y}

\text{RHS}

= \frac{1}{(\log x / \log z)} + \frac{\log y}{\log z}

= \frac{1}{(\log x + \log y)/\log z}

= \frac{\log z}{\log x + \log y}

Phew!

I'm not sure what you would use it for, though!

Regards

## Re: Log Laws

Thanks. I doubt it has a use, too, but I found it interesting.

X

Thanks. I doubt it has a use, too, but I found it interesting.