4.3 Logarithmic Functions
Ideas in this lesson
- Use the inverse definition and evaluate logarithms
- Build logarithm graphs and recover their bases
- Transform logarithms and distinguish absolute values
- Determine real domains of composite logarithms
- Analyze all branches of nonlinear logarithmic graphs
- Compose exponential and logarithmic functions in both orders
- Compare logarithmic growth and parameter families
- Invert nested logarithms, towers and logistic quotients
- Interpret supplied logarithmic models with units
- Prove large-scale facts and derive logarithmic secants
Use the inverse definition and evaluate logarithms · Worked example
Complete the exact common-log table for arguments \(\dfrac{1}{1000}\), \(\dfrac{1}{100}\), \(\dfrac{1}{10}\), \(1\), \(\sqrt{10}\), \(10\), \(100\), \(1000\). Convert \(5^u=v\) and \(\log_{5}(3z)=w\) into the other form, stating argument restrictions. Convert each of these six given statements into its other form and reverse the conversion to check it:
Also evaluate \(e^{4\ln(5)}\) exactly using an inverse identity. Evaluate \(\log_{5}(\sqrt{5})\) exactly and convert it to exponential form. State and explain both inverse identities and their domains. Find the valid base in \(\log_a(625)=4\) and rewrite \(\ln u=0\) exponentially. Define common/natural logarithms and evaluate \(\dfrac{\ln{\left(\dfrac{5}{2} \right)}}{\ln{\left(10 \right)}}\), \(\ln{\left(2 \right)}\), \(\ln{\left(1 + \sqrt{5} \right)}\) to four decimals. Show the requested work; written explanations and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
| Row | Given statement | Required conversion |
|---|---|---|
| 1 | \(\log_{5}(25)=2\) | Write the equivalent form |
| 2 | \(6^{-1}=\dfrac{1}{6}\) | Write the equivalent form |
| 3 | \(\log_{10}(1)=0\) | Write the equivalent form |
| 4 | \(4^{\dfrac{3}{2}}=8\) | Write the equivalent form |
| 5 | \(\log_{5}(\sqrt{5})=\dfrac{1}{2}\) | Write the equivalent form |
| 6 | \(4^{- \dfrac{3}{2}}=\dfrac{1}{8}\) | Write the equivalent form |
Answer: \(\dfrac{1}{2}\); valid base \(5\); \(u=e^{0}\).
\(\log_a u=w\iff a^w=u\) requires \(a>0,\ a\ne1,\ u>0\). The base stays fixed; the logarithm gives the exponent. The exponential is one-to-one, so its inverse exists.
\(\log_{5}(\sqrt{5})=\dfrac{1}{2}\) because \(5^{\dfrac{1}{2}}=\sqrt{5}\).
\(\log_{5}(5^{x})=x\) holds for every real input, whereas \(5^{\log_{5}x}=x\) requires \(x>0\).
\(\log_a(625)=4\iff a^{4}=625\). Raising the positive argument to the reciprocal power gives \(a=(625)^{\dfrac{1}{4}}=5\). Retain only bases \(a>0,\ a\ne1\).
\(\ln u=0\iff u=e^{0}\). The natural logarithm has base \(e\); the common logarithm has base 10.
Numerical evaluations, with the full argument grouped before rounding: \(\dfrac{\ln{\left(\dfrac{5}{2} \right)}}{\ln{\left(10 \right)}}\approx0.3979\), \(\ln{\left(2 \right)}\approx0.6931\), \(\ln{\left(1 + \sqrt{5} \right)}\approx1.1744\).
\(5^u=v\) is equivalent to \(\log_{5}v=u\) with \(v>0\). Also \(\log_{5}(3z)=w\) is equivalent to \(3z=5^w\), requiring \(z>0\). These symbolic conversions preserve the whole argument, not just its variable.
The exact common-log table follows from \(\log_{10}(10^j)=j\).
\(e^{4\ln(5)}=(e^{\ln(5)})^{4}=5^{4}=625\). The positive argument permits the inverse identity before applying the integer power.
Written-work review:
\(\log_{5}(\sqrt{5})=\dfrac{1}{2}\) because \(5^{\dfrac{1}{2}}=\sqrt{5}\).
\(\log_{5}(5^{x})=x\) holds for every real input, whereas \(5^{\log_{5}x}=x\) requires \(x>0\).
\(\log_a(625)=4\iff a^{4}=625\). Raising the positive argument to the reciprocal power gives \(a=(625)^{\dfrac{1}{4}}=5\). Retain only bases \(a>0,\ a\ne1\).
\(\ln u=0\iff u=e^{0}\). The natural logarithm has base \(e\); the common logarithm has base 10.
Numerical evaluations, with the full argument grouped before rounding: \(\dfrac{\ln{\left(\dfrac{5}{2} \right)}}{\ln{\left(10 \right)}}\approx0.3979\), \(\ln{\left(2 \right)}\approx0.6931\), \(\ln{\left(1 + \sqrt{5} \right)}\approx1.1744\).
\(5^u=v\) is equivalent to \(\log_{5}v=u\) with \(v>0\). Also \(\log_{5}(3z)=w\) is equivalent to \(3z=5^w\), requiring \(z>0\). These symbolic conversions preserve the whole argument, not just its variable.
The exact common-log table follows from \(\log_{10}(10^j)=j\).
| Argument | Common logarithm |
|---|---|
| \(\dfrac{1}{1000}\) | \(-3\) |
| \(\dfrac{1}{100}\) | \(-2\) |
| \(\dfrac{1}{10}\) | \(-1\) |
| \(1\) | \(0\) |
| \(\sqrt{10}\) | \(\dfrac{1}{2}\) |
| \(10\) | \(1\) |
| \(100\) | \(2\) |
| \(1000\) | \(3\) |
| Row | Logarithmic form | Equivalent exponential form |
|---|---|---|
| 1 | \(\log_{5}(25)=2\) | \(5^{2}=25\) |
| 2 | \(\log_{6}(\dfrac{1}{6})=-1\) | \(6^{-1}=\dfrac{1}{6}\) |
| 3 | \(\log_{10}(1)=0\) | \(10^{0}=1\) |
| 4 | \(\log_{4}(8)=\dfrac{3}{2}\) | \(4^{\dfrac{3}{2}}=8\) |
| 5 | \(\log_{5}(\sqrt{5})=\dfrac{1}{2}\) | \(5^{\dfrac{1}{2}}=\sqrt{5}\) |
| 6 | \(\log_{4}(\dfrac{1}{8})=- \dfrac{3}{2}\) | \(4^{- \dfrac{3}{2}}=\dfrac{1}{8}\) |
\(e^{4\ln(5)}=(e^{\ln(5)})^{4}=5^{4}=625\). The positive argument permits the inverse identity before applying the integer power.
Written-work review:
- Complete all eight exact common-log values and all six bidirectional log/exponential conversion rows.
- Preserve the base, exponent and positive argument and state the unequal domains of the two inverse identities.
- Solve the unknown base/argument while rejecting invalid bases and evaluate the scaled-log inverse expression exactly.
- Evaluate the separate common/natural logarithms with grouped arguments and four-decimal rounding.