# Log Laws [Solved!]

**MichaelA** 15 Dec 2015, 02:48

### 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

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

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

**Newton** 15 Dec 2015, 15:10

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

**MichaelA** 16 Dec 2015, 05:48

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.

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