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Here is one way of looking at it. I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$: $$ \tag{$\spadesuit$}[b \to x] = [b \to a][a \to x] $$ or equivalently $$ [a \to x] = [b \to x]/[b \to a]. $$ In standard notation, this formula states that $$ \log_a(x) = \frac{\log_b(x)}{\log_b(a)}. $$

Note that equation $(\spadesuit)$ is obvious, because \begin{align} b^{[b \to a][a \to x]} &=(b^{[b\to a]})^{[a \to x]} \\ &= a^{[a \to x]} \\ &= x. \end{align}

Here is one way of looking at it. (I'll assume that the numbers $a,b,x \in \mathbb R$ satisfy $a > 1, b > 1$, and $x > 0$.)

I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$: $$ \tag{$\spadesuit$}[b \to x] = [b \to a][a \to x] $$ or equivalently $$ [a \to x] = [b \to x]/[b \to a]. $$ In standard notation, this formula states that $$ \log_a(x) = \frac{\log_b(x)}{\log_b(a)}. $$

Note that equation $(\spadesuit)$ is obvious, because \begin{align} b^{[b \to a][a \to x]} &=(b^{[b\to a]})^{[a \to x]} \\ &= a^{[a \to x]} \\ &= x. \end{align}

Here is one way of looking at it. I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$: $$ [b \to x] = [b \to a][a \to x] $$ or equivalently $$ [a \to x] = [b \to x]/[b \to a]. $$ In standard notation, this formula states that $$ \log_a(x) = \frac{\log_b(x)}{\log_b(a)}. $$

Here is one way of looking at it. I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$: $$ \tag{$\spadesuit$}[b \to x] = [b \to a][a \to x] $$ or equivalently $$ [a \to x] = [b \to x]/[b \to a]. $$ In standard notation, this formula states that $$ \log_a(x) = \frac{\log_b(x)}{\log_b(a)}. $$

Note that equation $(\spadesuit)$ is obvious, because \begin{align} b^{[b \to a][a \to x]} &=(b^{[b\to a]})^{[a \to x]} \\ &= a^{[a \to x]} \\ &= x. \end{align}

I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ might be "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The exponent rule tells us that the exponent from $b$ to $y$ is equal to the exponent from $b$ to $x$ multiplied by the exponent from $x$ to $y$: $$ [b \to y] = [b \to x][x \to y]. $$

Here is one way of looking at it. I dislike the name "logarithm" and I think a more descriptive name for $\log_b(x)$ is "the exponent from $b$ to $x$". We could also use the notation $[b \to x]$ instead of $\log_b(x)$. The change of base rule then tells us that the exponent from $b$ to $x$ is equal to the exponent from $b$ to $a$ times the exponent from $a$ to $x$: $$ [b \to x] = [b \to a][a \to x] $$ or equivalently $$ [a \to x] = [b \to x]/[b \to a]. $$ In standard notation, this formula states that $$ \log_a(x) = \frac{\log_b(x)}{\log_b(a)}. $$