Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation \(log_b(x)=y\) is equivalent to the exponential equation \(b^y=x\). We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation \(log_2(2)+log_2(3x−5)=3\).  To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for 5\(x\):

\(log_2(2)+log_2(3x−5)=3\)
\(log_2(2(3x−5))=3\)
Apply the product rule of logarithms.
\(log_2(6x−10)=3\)
Distribute.
\(2^3=6x−10\)
Apply the definition of a logarithm.
\(8=6x−10\)
Calculate \(2^3\).
\(18=6x\)
Add 10 to both sides.
\(x=3\)
Divide by 6.

Using the Definition of a Logarithm to Solve Logarithmic Equations

For any algebraic expression \(S\) and real numbers \(b\) and \(c\), where \(b > 0\) , \(b≠1\),

\(log_b(S)=c\) if and only if \(b^c=S\)

Example 9

Using Algebra to Solve a Logarithmic Equation

Solve \(2lnx+3=7\).

Solution

\(2lnx+3=7\)
\(2lnx=4\)
Subtract 3.
\(lnx=2\)
Divide by 2.
\(x=e^2\)
Rewrite in exponential form.

Try It #9

Solve \(6+lnx=10\).


Example 10

Using Algebra Before and After Using the Definition of the Natural Logarithm

Solve \(2ln(6x)=7\).

Solution

\(2ln(6x)=7\)
\(In (6x) = \dfrac{7}{2}\)
Divide by 2.
\(6x = e^{(\dfrac{7}{2})}\)
Use the definition of \(ln\).
\(x = \dfrac{1}{6} e^{(\dfrac{7}{2})}\)
Divide by 6.


Try It #10

Solve \(2ln(x+1)=10\).


Example 11

Using a Graph to Understand the Solution to a Logarithmic Equation

Solve \(ln x=3\).

Solution

\(ln x=3 \)

\(x=e^3\)  Use the definition of the natural logarithm.

Figure 3 represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words \(e^3≈20\). A calculator gives a better approximation: \(e^3≈20.0855\).

Graph of two questions, y=3 and y=ln(x), which intersect at the point (e^3, 3) which is approximately (20.0855, 3).


Figure 3 The graphs of \(y=ln x\) and \(y=3\) cross at the point \((e^3,3)\), which is approximately (20.0855, 3).

Try It #11

Use a graphing calculator to estimate the approximate solution to the logarithmic equation \(2^x=1000\) to 2 decimal places.

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-6-exponential-and-logarithmic-equations
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