The logarithm is used for finding of an exponent in which it is necessary to build the basis for receiving the number specified under the sign of a logarithm. Not necessarily under the sign of a logarithm there has to be a number - it is possible to specify a variable, a polynomial, function, etc. May contain podlogarifmenny expression and one more logarithm. Operation of calculation of a logarithm from a logarithm of special complexity does not represent especially as often it can be simplified transformations of an internal logarithm.
Instruction
1. Finding of a logarithm in itself from a logarithm does not assume any special transformations - just execute consistently two such operations. The only feature - it is necessary to begin with an internal logarithm, i.e. with that which is podlogarifmenny expression of another. For example, if it is necessary to find log ₃ log ₂ 512, begin with calculation of a logarithm 512 on the basis 2 (log ₂ 512 = 9), and then count a logarithm of the received result with the basis 3 (log ₃ 9 = 2), i.e. log ₃ log ₂ 512 = log ₃ 9 = 2.
2. If one of podlogarifmenny expressions is the polynomial, use transformation formulas before starting calculations. For example, transform the sum of logarithms on the identical basis to a logarithm of performing their podlogarifmenny expressions on the same basis: log ₐ (log ᵤ x + log ᵤ y) = log ₐ log ᵤ (x*y). In the similar way transform also the difference of logarithms: log ₐ (log ᵤ x - log ᵤ y) = log ₐ log ᵤ (x/y).
3. In certain cases, if podlogarifmenny expression contains number or a variable built in degree there is an opportunity even more to simplify expression. Let's tell, the example of log used in the first step ₃ log ₂ 512 can be presented in such form: log ₃ log ₂ 2 ⁹. It allows to bring 9 out of the sign of an internal logarithm and need to calculate a logarithm 512 will disappear as log ₃ log ₂ 2 ⁹ = log ₃ (9*log ₂ 2) = log ₃ (9*1) = 2.
4. The rule described in the previous step can be applied also to logarithms from the expressions containing a root or fraction. For this purpose present a root in the form of a fractional exponent of degree. For example, if it is necessary to find log ₃ log ₂ ⁹√ 2, then ⁹√ 2 it is possible to present as 2 to degrees 1/9. Then log2 ⁹√ 2 = 1/9 * log ₂ 2 = 1/9 = 1/3² = 3 ⁻². And log ₃ 3 ⁻² =-2. All these transformations allowed to do in general without calculations, and it is possible to write down the decision so: log ₃ log ₂ ⁹√ 2 = log ₃ (1/9 * log ₂ 2) = log ₃ (1/9) = log ₃ (1/3²) = log ₃ 3 ⁻² =-2.