Section 1.1 Calculus 2 - Professor Clontz

1.1 Logarithms and Exponential Functions

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 \(\int( \frac{3}{x^2}-\frac{2}{x}+1+4x^2)\,dx\).

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 \(e\approx 2.718\).

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.)

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.)
Find \(\int( \frac{6}{x^3}+\frac{2}{x}-3x)\,dx\).
Find \(\int \frac{6x^4-x^2+4}{2x^3}\,dx\).
We saw that \(\frac{d}{dx}[e^x]=e^x\). Describe infinitely many other functions \(f(x)\) such that \(f’(x)=f(x)\).
Find \(\frac{d}{dx}[\frac{1}{x}+3e^x]\).
Prove the following derivative formulas: \( \frac{d}{dx}[\log_b x]=\frac{1}{x\ln b} \) and \( \frac{d}{dx}[a^x]=a^x \ln a \).
Find \(\int (3x^4+3e^x-\frac{4}{x})\,dx\).
Differentiate \(f(x)=\ln(x^2)+e^{x^3}\).