# Section 1.1 Calculus 2

Logarithms and Exponential Functions

## 1.1 Logarithms and Exponential Functions

### 1.1.1 The Natural Logarithm

• Using integrals, we may rigorously define a logarithm.
• $$\ln x=\int_1^x \frac{1}{t}\,dt$$ for all $$x>0$$
• $$\frac{d}{dx}[\ln x] = \frac{1}{x}$$ and $$\ln 1 = 0$$
• Example. Use the definition $$\ln x=\int_1^x \frac{1}{t}\,dt$$ to prove the property $$\ln(ax) = \ln a + \ln x$$ for $$a,x>0$$. (Hint: start by showing that the derivatives are the same.)
• This allows us to express an indefinite integral for $$1/x$$: $$\int\frac{1}{x}\,dx=\ln|x|+C$$. (Note the absolute value.)
• Example. Find $$\int( \frac{3}{x^2}-\frac{2}{x}+1+4x^2)\,dx$$.

### 1.1.2 The Natural Number and Natural Exponential Function

• Note that $$a^p$$ has only been defined for when $$p\in\mathbb Q$$.
• Since $$f(x)=\ln x$$ is differentiable and 1-to-1, we can define $$\exp x=f^{\leftarrow}(x)$$ to be its differentiable inverse.
• Example. Use the fact $$\frac{d}{dx}[f^{\leftarrow}(x)]=\frac{1}{f’(f^{\leftarrow}(x))}$$ to prove that $$\frac{d}{dx}[\exp x]=\exp x$$. (Hint: let $$f(x)=\ln x,f’(x)=\frac{1}{x},f^{\leftarrow}(x)=\exp x$$.)
• Let $$e=\exp 1$$. We’ll see much later in the course why $$e\approx 2.718$$.

### 1.1.3 General Logarithms and Exponential Functions

• Since $$\exp x$$ is defined for all real numbers, we may define $$a^x = \exp(x\ln a)$$ for all $$a>0$$ and $$x\in\mathbb R$$. Note that $$e^x = \exp x$$.
• Example. Use the definition $$a^x = \exp(x\ln a)$$ and property $$\ln(abc)=\ln a + \ln b + \ln c$$ to show that $$2^3 = 2\times2\times2$$.
• Define $$\log_b x = \frac{\ln x}{\ln b}$$ for $$b>1$$.
• Example. Use the definitions $$\log_b x = \frac{\ln x}{\ln b}$$ and $$b^x = \exp(x\ln b)$$ to prove the property $$x = \log_b(b^x)$$. (That is, $$\log_b x$$ and $$b^x$$ are inverse functions.)

### Review Exercises

1. Use the definition $$\ln x=\int_1^x \frac{1}{t}\,dt$$ to prove the property $$\ln(x^p) = p\ln x$$ for $$x>0$$ and $$p\in\mathbb Q$$. (Hint: start by showing that both sides share the same derivative.)
2. Find $$\int( \frac{6}{x^3}+\frac{2}{x}-3x)\,dx$$.
3. Find $$\int \frac{6x^4-x^2+4}{2x^3}\,dx$$.
4. We saw that $$\frac{d}{dx}[e^x]=e^x$$. Describe infinitely many other functions $$f(x)$$ such that $$f’(x)=f(x)$$.
5. Find $$\frac{d}{dx}[\frac{1}{x}+3e^x]$$.
6. Prove the following derivative formulas: $$\frac{d}{dx}[\log_b x]=\frac{1}{x\ln b}$$ and $$\frac{d}{dx}[a^x]=a^x \ln a$$.
7. Find $$\int (3x^4+3e^x-\frac{4}{x})\,dx$$.
8. Differentiate $$f(x)=\ln(x^2)+e^{x^3}$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 7.1 (review: 1.5,1.6)