# Find the exact value of the logarithmic expression `log_3 45 - log_3 15` without using a calculator.

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You may use the alternative method to solve this problem without the presence of calculator.

You should try to express 45 as the product `3*15` , hence, taking logarithm of 45 yields:

`log_3 45 = log_3 (3*15)`

You should expand the logarithm of product into a sum of two logarithms of factors such that:

`log_3 (3*15) = log_3 3 + log_3 15`

Hence, substituting `log_3 3 + log_3 15` for `log_3 45` yields:

`log_3 3 + log_3 15 - log_3 15`

You need to reduce like terms such that:

`log_3 3 + log_3 15 - log_3 15`

`log_3 3 + log_3 15 - log_3 15 = log_3 3 `

You need to remember that `log_3 3 = 1` , hence `log_3 45 - log_3 15 = 1` .

You may also use the factorization even more such that:

`15 = 3*5`

`45 = 3*3*5`

Hence, `log_3 45 - log_3 15 =log_3(3*3*5) - log_3 (3*5) = log_3 3 + log_3 3 + log_35- log_3 3- log_3 5`

Reducing like terms yields `log_3 45 - log_3 15 = 1.`

**Hence, you may use alternative methods to calculate the difference of logarithms, without the presence of calculator.**

The value of the logarithm `log_3 45 - log_3 15` has to be determined.

Use the property log a - log b = log(a/b)

`log_3 45 - log_3 15`

=> `log_3(45/15)`

=> `log_3 3`

Use the property `log_a a = 1`

=> 1

**The value of **`log_3 45 - log_3 15 = 1`