16.5 Derivatives of Logarithmic Function
\[\boxed {\frac {d}{dx}\, \log _a (x)=\frac {1}{x\, ln (a)}}\]
We can verify this differentiation formula.
Let \(\,m=\log _a (x)\, \iff \, a^m = x\) \[\boxed {a^m = x\quad \cdots \quad (1)}\]
Now take the natural log on both sides of equation (1) \begin {align*} \ln (a^m) & = \ln (x)\\\\ \implies \quad m\, \ln (a) & = \ln (x)\\\\ \implies \quad m & = \frac {\ln (x)}{\ln (a)} \end {align*}
Therefore, \(\displaystyle {\log _a(x)= m = \frac {\ln (x)}{\ln (a)}}\)
Hence \(\displaystyle {\frac {d}{dx}\, \log _a(x) = \frac {1}{\ln (a)}\,\frac {d}{dx}\, \ln (x)}\)
\[\implies \quad \frac {d}{dx}\, \log _a(x) = \frac {1}{x\, \ln (a)}\]
Therefore, the general formula is \begin {align*} \frac {d}{dx}\, \log _a[f(x)] & = \frac {1}{\ln (a)}\, \frac {d}{dx}\, \ln [f(x)]\\\\ \implies \quad \frac {d}{dx}\, \log _a[f(x)] & = \frac {f'(x)}{f(x)\, \ln (a)}\\\\ \end {align*}
Solution.
\(y = \log _{10}(x^3 + 2x^2)\)
Let \(f(x) = x^3 + 2x^2\, \implies \, f'(x) = 3x^2 + 4x\)
\begin {align*} \frac {d}{dx}\, \log _a [f(x)] & = \frac {f'(x)}{f(x) \, \ln (a)}\\\\ \implies \, \frac {d}{dx}\, \log _{10}(x^3 + 2x^2) & = \frac {3x^2 + 4x}{(x^3 + 2x^2)\, \ln (10)}\\\\ \therefore \, \frac {d}{dx}\, \log _{10}(x^3 + 2x^2) & = \frac {3x + 4}{(x^2 + 2x)\, \ln (10)}\\\\ \end {align*}
Sample Questions
- 1.
- Change \(\log _38\sqrt {2}\) to base 2.
- 2.
- Find \(x\) if \(\log 32=x\).
- 3.
- Simplify \(\log _a4+2\log _a3-\log _a\).
- 4.
- Express \(\log _a\frac {x^3}{y^2z}\) in terms of \(\log _ax\), \(\log _ay\), \(\log _az\).
- 5.
- \(3^{2x}-6(3^x)+5=0\)
- 6.
- \(2(5^{2x})-5^x=6\)
- 7.
- \(6^{2x-1}=9^{x+3}\)
- 8.
- \(2^{2x+1}=3(2^x)-1\)
- 9.
- \(\log _3x-2\log _x3=1\)
- 10.
- If \(xy=64\) and \(\log _xy+\log _yx=\frac {5}{2}\). Find \(x\) and \(y\).
- 11.
- Solve for \(x\) and \(y\)
- (a)
- \begin {align*} \log 2+\log x & = 2\log y\\ 3x-4y-8 & = 0 \end {align*}
- (b)
- \begin {align*} \log _2(x-3y+2) & =0\\ \log _2(x+1)-1 & = 2\log _2y \end {align*}
- 12.
- Solve the equation \(\log _2\sqrt {x}=\sqrt {\log _2x}\).
- 13.
- Solve the systems of equations \[\log _xy=2,\quad xy=8\]
- 14.
- Solve the equation \(\log _x4=\frac {\log _28}{x\log _4x}\).
- 15.
- Given that \(\log _2x+\frac {\log _7y}{\log _72}=\log _381\), express \(y\) in terms of \(x\).
- 16.
- Let \(y=f(x)=-\log _2(1-x)\)
- (a)
- state the domain and range of \(y=f(x)\)
- (b)
- state the asymptote of \(y=f(x)\)
- (c)
- sketch the graph of \(y=-\log _2(1-x)\).
- 17.
- Show that \[4\log x-3\log (x^2+1)+2\log (x-1) = \frac {\ln \left [\frac {x^4(x-1)^2}{(x^2+1)^3}\right ]}{\ln 10}\]
- 18.
- Solve the equation \[x^2 (10)^x-x(10)^x=2(10)^x\]
- 19.
- Find the derivatives of each function below:
- (a)
- \(g(x)=\ln \left (\frac {1-x}{1+x}\right )\)
- (b)
- \(f(x)=2^{\displaystyle {-x}}+\cos \left (\frac {1}{x}\right )\)
- 20.
- Find \(\frac {dy}{dx}\)
- (a)
- \(y=\frac {\sqrt {x}-1}{\sqrt {x}+1}\)
- (b)
- \(y=2^{\displaystyle {\cos x^{\displaystyle {2}}}}\)
- 21.
- Given \(f(x)=\log _{2}x\) and \(h(x)=\log _{\frac {1}{2}}(x-1)\)
- (a)
- sketch, on the same graph, the function \(f(x)\) and \(h(x)\).
- (b)
- find the value of \(x\) for the point where the graphs intersect.
- 22.
- Solve the following equations
- (a)
- \(4x+8=33\sqrt {x}\)
- (b)
- \(e^{3x}-3e^{2x}-e^x+3=0\)
- 23.
- Solve for \(x\) given that, \(6x^{-3/2}=\frac {3}{32}\)
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