Learn how to solve elementary equations containing exponential functions based on definition. Practice logarithmic functions calculus and properties of log x graph functions.
For \(a>1\), the graph of \(log_ax\) for different values of \('a'\) will be as shown in figure.
Let \(f(x)=a^x\), we define \(f^{-1}(x)=log_a\,x\) as the inverse function of \(f\). It is called the logarithmic function with base \('a'\).
\(log_a\,x=y\;\iff\;a^y=x\)
A single logarithmic function which contain complicated expression can be simplified to many \(log\) term by using the above properties.
Example: \(log_{10}\,3\sqrt{\dfrac{y+3}{2x+7}}\) can be written as,
\(=log_{10}\left(\dfrac{y+3}{2x+7}\right)^{1/3}\)
\(=\dfrac{1}{3}log_{10}\left(\dfrac{y+3}{2x+7}\right)\)
\(=\dfrac{1}{3}\left[log_{10}(y+3)-log_{10}(2x+7)\right]\)
A \(\dfrac{1}{2}[log_{10}\,(x^2+1)-log_{10}\,(x^2+3)]\)
B \(\dfrac{1}{2}log\,\left(\dfrac{x^2+3}{x^2+1}\right)\)
C \(\dfrac{1}{2}[log_{10}\,(x^2+4)-log_{10}\,x]\)
D \(2\,log\,x^2\)
If we need to solve a simple logarithmic equation often, the definition of \(log\) will be required to convert that equation into simple linear one.
Example: Solve \(log_{2}\,(3x-5)=3\)
\(\Rightarrow\,3x-5=2^3\) \((\log_{a}\,x=y\Rightarrow\,x=a^y)\)
\(\Rightarrow\,3x=5+8\)
\(\Rightarrow\,x=\dfrac{13}{3}\)
A \(x=\dfrac{102}{5}\)
B \(x=\dfrac{5}{7}\)
C \(x=-\dfrac{2}{3}\)
D \(x=\dfrac{2}{105}\)
We can write, \(log_{10}\,6=log_{10}\,2×3=log_{10}\,2+log_{10}\,3\)
and \(log_{10}\dfrac{10}{3}=log_{10}\,10-log_{10}\,3=1-log_{10}\,3\)
and \(log_{10}\,25=log_{10}\,5^2=2log_{10}\,5\)
Sometimes the reverse of above properties will be used in the problem.
e.g.
\(5\,log_{10}\,x^2-2\,log\,y+7\,log\dfrac{5}{y}\)
can be written as,
\(log_{10}\,x^{10}-log\,y^2+log\,\left(\dfrac{5}{y}\right)^7\)
\(=log_{10}\,\dfrac{x^{10}}{y^2}×\left(\dfrac{5}{y}\right)^7\)
\(=log_{10}\,\dfrac{x^{10}\,5^7}{y^9}\)
A \(log_{10}\,(x^2y^3)\)
B \(log_{10}\,\left(\dfrac{y^2}{x^3}\right)\)
C \(log_{10}\left(\dfrac{x^4}{y^5}\right)\)
D \(log_{10}\left(\dfrac{y^5}{x^4}\right)\)
If the given equation contains exponential function in which the exponents contain variable \(x\) then to solve for \(x\), we take \(log\) on both sides to the same base whose power is raised. This will convert the equation to simple equation.
e.g. Solve \(3^{2x-7}=5\)
Take \(log\) , both sides to base 3
\(\Rightarrow\,(2x-7)\log_{3}\,3=\log_{3}\,5\)
\(\Rightarrow\,2x-7=\log_{3}\,5\)
\(\Rightarrow\,x=\dfrac{7+\log_{3}\,5}{2}\)
A \(x=\dfrac{(\log_{5}\,4+3)}{2}\)
B \(x=\log_{4}\,5\)
C \(x=\dfrac{(log_{4}\,5+11)}{9}\)
D \(x=e^2\)
\(ln\) stands for \(log\) natural.
A \(x=ln\,2\)
B \(x=ln\,3\)
C \(x=ln\,6\)
D \(x=ln\,500\)