Logarithm problem 2

Let $x$ be any real number and let $a, b, n$ be positive real number with $a \not = 1$ and $b \not = 1$ . Show that if $x = \log_{a}n$ then $x = \displaystyle{\frac{\log_{b}n}{\log_{b}a}}$.

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

By definition of logs:- $x = \log_{a}n\implies a^x=n$ Taking $log$ with base $b$ on both sides:- $\log_b a^x=\log_b n$ $\implies x \log_b a=\log_b n$ $\implies x=\dfrac{\log_b n}{\log_b a}$

Rishabh Jain - 5 years, 3 months ago

$\begin{aligned} x & =\log_{a}n = \frac{\log n}{\log a} \\ & = \frac{ \frac{\log n}{\log b}}{\frac{\log a}{\log b}} = \frac{\log_{b} n}{\log_{b}a}\end{aligned}$

Akshat Sharda - 5 years, 3 months ago

Well that's basically applying what we need to prove with b=e.

Vishnu Bhagyanath - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$