Формула заміни основи

The Change of Base Formula

Let's say you have a somewhat mediocre calculator; one that can only calculate logs in base $10$.

Now let's think about how we would approximate something like $\log_5{3}$. Before we touch the calculator, let's try to manipulate it so that we can get something we can calculate.

Let's raise that to a base of $5$:

$$5^{\log_5{3}}=3$$

Now we introduce something we can calculate: a log base $10$.

$$(10^{\log_{10}{5}})^{\log_5{3}}=3$$

We have something we can use! At this point, we attempt to isolate $\log_5{3}$.

$$10^{(\log_{10}{5})(\log_5{3})}=3$$ $$(\log_{10}{5})(\log_5{3})=\log_{10}{3}$$ $$\log_5{3}=\dfrac{\log_{10}{3}}{\log_{10}{5}}$$

And we've done it!

If we do the same for any $\log_b{a}$, we get $$\log_b{a}=\dfrac{\log{a}}{\log{b}}$$

And you can even change it into whatever base you want, so this is valid: $$\log_b{a}=\dfrac{\log_{427}{a}}{\log_{427}{b}}=\dfrac{\ln{a}}{\ln{b}}$$

(Video) Change of Base Formula

by mathman1024

Here's a video by mathman1024 going over the Change of Base Theorem.

The common logarithm is in base $10$. When writing the common logarithm we leave out the ten but it's implied that it's there. Both sides of this equation are equal. $$\log x=\log_{10}x$$ Just like how when we write just the radical symbol, it's implied that there's a two.

The natural logarithm has a base of $e$. We denote this by writing $\ln$ instead of $\log$. $$\ln x=\log_ex$$

The change of base formula is used to convert a logarithm to a different base. Pay close attention to the numerator and denominator. It says: $$\log_ax=\frac{\log_bx}{\log_ba}$$

The first example he looks at is: $\log_5125$. He converts it into base 10 so he can input it into his calculator. Apply the change of base formula. $$\begin{align} \log_5125&=\frac{\log125}{\log5}\\ &=3 \end{align}$$

Next: $\log_{32}8$. $$\begin{align} \log_{32}8&=\frac{\log_28}{\log_232}\\ &=\frac32 \end{align}$$

Now that we've learned how to add and subtract logs, we can move onto more complicated math with logs.

Change of Base Formula for Logs

We know that logarithms can have many different bases, such as $2$, $10, or even$e$. However, most calculators only have buttons for common logs (base$10$) and natural logs (base$e$).

The change of base formula lets you rewrite a log with any base into an equivalent log using a base your calculator can evaluate.

If $b$, $a$, and $x$ are positive numbers with $b\neq1$ and $a\neq1$, then:

$$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$$

We have two fairly common versions for both common logs and natural logs.

For common logs:

$$\log_b(x) = \frac{\log(x)}{\log(b)}$$

For natural logs:

$$\log_b(x) = \frac{\ln(x)}{\ln(b)}$$

Both formulas will give you the exact same answer.

####Example

Let's say you want to evaluate $\log_7(25$.

Most calculators don't have a button for $\log_7$. This is where the change of base formula comes in.

$$\log_7(25) = \frac{\log(25)}{\log(7)}$$

We can enter that into the calculator to find that:

$$\log_7(25) = 1.654$$

Do note that it does not matter whether you use a common logarithm or a natural logarithm. For any logarithm,

$$\frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}$$

Both expressions produce the same value because they are equivalent forms of the change of base formula.

####Solve Exponential Equations

The change of base formula is especially useful when solving exponential equations.

Let's say that $2^x = 15$.

We can rewrite the equation using logarithms: $x=\log_2(15)$.

We can now use the change of base formula!

$$x=\frac{\log(15)}{\log(2)}$$

We can solve, and we'll find that $x=3.907$.