4.3 Logarithmic functions and laws

Logarithms answer exponent questions and provide the inverse operation needed to solve many exponential equations.

4.3.1 Definition, graph, and domain

Logarithm. For \(b>0\) and \(b\ne1\), \(\log_b(y)=x\) exactly when \(b^x=y\).

The logarithm’s input must be positive. The graph of \(y=\log_bx\) has domain \((0,\infty)\), range \(\mathbb R\), \(x\)-intercept \((1,0)\), and vertical asymptote \(x=0\). It is the reflection of \(y=b^x\) across \(y=x\). Natural logarithm uses base \(e\) and is written \(\ln x\).

Worked Example: Converting forms

The statement \(4^{1/2}=2\) is equivalent to \(\log_4(2)=1/2\). Likewise, \(\log_3(1/9)=-2\) because \(3^{-2}=1/9\).

Try It. Rewrite \(5^3=125\) in logarithmic form.
Check Your Work \(\log_5(125)=3\).

Worked Example: Finding a logarithmic domain

For \(f(x)=\log_7(9-5x)\), require \(9-5x>0\). Therefore, \(x<9/5\).

Try It. Find the domain of \(\ln(3x+12)\).
Check Your Work \(3x+12>0\), so \(x>-4\).

4.3.2 Product, quotient, and power laws

For positive arguments,

\[\log_b(MN)=\log_bM+\log_bN,\] \[\log_b(M/N)=\log_bM-\log_bN,\] \[\log_b(M^p)=p\log_bM.\]

These laws do not permit splitting a sum or difference inside a logarithm. Domains must be taken from the original expression.

Worked Example: Expanding a logarithm

For positive \(x\) and \(x+1\),

\[\ln\left(\frac{3e^x(x+1)^5}{x^2}\right)=\ln3+x+5\ln(x+1)-2\ln x.\]

Try It. Expand \(\log_2[8x^3/(x-1)]\).
Check Your Work \(3+3\log_2x-\log_2(x-1)\), on the original domain.

Worked Example: Combining logarithms

\[3\ln x+2\ln(x+1)-\ln(4x-1)=\ln\left(\frac{x^3(x+1)^2}{4x-1}\right).\]

Try It. Combine \(2\log_5x-\log_5(x+3)\).
Check Your Work \(\log_5[x^2/(x+3)]\).

Applied Problem: Determining a pressure parameter. A model gives \(\ln(k)=-21.504\), where \(k\) is measured in Pa\(^{-1}\). Solve for \(k\) and report it in scientific notation.

Check Your Work Apply the exponential function: \(k=e^{-21.504}\approx4.59\times10^{-10}\) Pa\(^{-1}\).

4.3.3 Practice Problems

  1. Rewrite \(2^{-4}=1/16\) logarithmically.
    Check Your Work \(\log_2(1/16)=-4\).
  2. Evaluate \(\log_{10}(1000)\).
    Check Your Work 3.
  3. Find the domain of \(\ln(7-2x)\).
    Check Your Work \(x<7/2\).
  4. Expand \(\ln(x^4/y^2)\).
    Check Your Work \(4\ln x-2\ln y\) for positive \(x,y\).
  5. Combine \(\log x+3\log y\).
    Check Your Work \(\log(xy^3)\).
  6. A water model gives \(\ln k=-8.2\). Find \(k\).
    Check Your Work \(k=e^{-8.2}\approx2.75\times10^{-4}\).
  7. Find the domain of \(\log(12-C)\) for concentration \(C\ge0\).
    Check Your Work \(0\le C<12\).
  8. Expand \(\ln[Q^2/(Q+5)]\) for positive quantities.
    Check Your Work \(2\ln Q-\ln(Q+5)\).