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Derivation of the Base Change Formula for Logarithms 📂Functions

Derivation of the Base Change Formula for Logarithms

Formula

For any positive number $c>0$, $$ \log_{a} b = {{ \log_{c} b } \over { \log_{c} a }} $$

Explanation

Nowadays, the formula itself has become less meaningful, but it still holds significant value for entrance exams. It is recommended to solve many practice problems suitable for the title of ‘formula’, rather than underestimating it simply because it appears to be a simple property.

Derivation

If we say $x := \log_{a} b$, according to the definition of logarithm, $$ a^x = b $$ Taking $\log_{c}$ on both sides, $$ \log_{c} a^x = \log_{c} b $$ By the property of logarithms, when we bring down $x$, $$ x \log_{c} a = \log_{c} b $$ Dividing both sides by $\log_{c} a$, $$ x = {{ \log_{c} b } \over { \log_{c} a }} $$ But since we initially said $x = \log_{a} b$, $$ \log_{a} b = {{ \log_{c} b } \over { \log_{c} a }} $$