Question 1

Do natural logarithms follow the same rules as log?

Accepted Answer

Yes, logarithms of all bases follow the same rules above. Recall that $ln(x)$ is simply a log with base e.

Question 2

What are the rules for natural log?

Accepted Answer

$$ln(ab)=ln(a)+ln(b)$$

$$ln(\frac{a}{b})=ln(a)-ln(b)$$

$$ln(b^c)=c ln(b)$$

$$ln(a)=\frac{log_b(a)}{log_b(e)}$$

Question 3

How do prove the product rule of the natural log

Accepted Answer

The main thing to understand is the log rules are a reflection of the exponent rules as they are inverse functions. First let's call $m=ln(a)$ and $n=ln(b)$. Putting these expressions in exponential form we have

$$a=e^m$$ and $$b=e^b$$

we can multiply a and b to get

$$ab=e^me^n$$ and applying the rule for multiplying exponents we get

$$ab=e^{m+n}$$

Taking the natural log of both sides

$$ln(ab)=ln(e^{m+n})$$

$$ln(ab)=m+n$$

Finally substituting our definitions of m and n we have

$$ln(ab)=ln(a)+ln(b)$$

And there you have it!

Question 4

How to prove the quotient rule for natural log

Accepted Answer

The main thing to understand is the log rules are a reflection of the exponent rules as they are inverse functions. First let's call $m=ln(a)$ and $n=ln(b)$. Putting these expressions in exponential form we have

$$a=e^m$$ and $$b=e^b$$

we can divide a and b to get

$$\frac{a}{b}=\frac{e^m}{e^n}$$ and applying the rule for multiplying exponents we get

$$\frac{a}{b}=e^{m-n}$$

Taking the natural log of both sides

$$ln(\frac{a}{b})=ln(e^{m-n})$$

$$ln(\frac{a}{b})=m-n$$

Finally substituting our definitions of m and n we have

$$ln(\frac{a}{b})=ln(a)-ln(b)$$

And there you have it!

Question 5

How to prove the power law for natural log

Accepted Answer

The main thing to understand is the log rules are a reflection of the exponent rules as they are inverse functions. First let's call $m=ln(a). Putting this expression into exponential form we have

$$a=e^m$$

we can raise a to the power of b to get

$$a^b=(e^m)^b$$ and applying the exponent rule for powers we get

$$a^b=e^{mb}$$

Taking the natural log of both sides

$$ln(a^b)=ln(e^{mb})$$

$$ln(a^b)=mb$$

Finally substituting our definition of m

$$ln(a^b)=bln(a)$$

And there you have it!

Question 6

How to prove the change of base rule for natural log

Accepted Answer

The main thing to understand is the log rules are a reflection of the exponent rules as they are inverse functions. First let's call $m=ln(a). Putting this expression into exponential form we have

$$a=e^m$$

Now lets take the logarithm of both sides with any base we like, let's call it c

$$log_c(a)=log_c(e^m)$$

Using the power rule we can write

$$log_c(a)=mlog_c(e)$$

Solving for m we have

$$\frac{log_c(a)}{log_c(e)}=m$$

Finally substituting our definition of m

$$ln(a)=\frac{log_c(a)}{log_c(e)}$$

And there you have it!

Mathematics & Physics Tutoring

Category: Physics

The Natural Logarithm and its Rules

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