How to Solve Natural Logarithms Problems? (+FREE Worksheet!)
How to solve natural logarithms problems, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
In this blog post, you will learn more about Natural Logarithms and how to solve problems related to natural logarithms.
Solve Natural Logarithms Problems: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Translate firstAsk: the base to what power gives the input?
- Use rules legallyProducts, quotients, and powers have rules; sums do not split.
- Protect the domainKeep the log input positive and track asymptotes when graphing.
Worked examples
Evaluate a log
- Ask 3 to what power equals 81.
- 3 to the fourth power is 81.
- The logarithm is that exponent.
Find a log domain
- The input is x – 5.
- Require x – 5 > 0.
- Solve the inequality.
Try one before moving on
Solve Natural Logarithms Problems: pop-up practice
Related Topics
Step by step guide to solve Natural Logarithms
- A natural logarithm is a logarithm that has a special base of the mathematical constant (e), which is an irrational number approximately equal to (2.71).
- The natural logarithm of (x) is generally written as ln (x), or (log_{e}{x}).
For education statistics and research
Natural Logarithms, Example 1:
Solve the equation for (x): (e^x=3)
Solution:
If (f(x)=g(x)),then: (ln(f(x))=ln(g(x))→ln(e^x)=ln(3) )
Use log rule: (log_{a}{x^b}=b log_{a}{x}), then: (ln(e^x)=x ln(e)→xln(e)=ln(3) )
(ln(e)=1), then: (x=ln(3) )
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Natural Logarithms, Example 2:
Solve equation for (x): (ln(2x-1)=1)
Solution:
Use log rule: (a=log_{b}{b^a}), then: (1=ln(e^1 )=ln(e)→ln(2x-1)=ln(e))
When the logs have the same base: (log_{b}{f(x)}=log_{b}{g(x)}), then: (f(x)=g(x))
then: (ln(2x-1)=ln(e)), then: (2x-1=e→x=frac{e+1}{2})
Natural Logarithms, Example 3:
Solve the equation for (x): (e^x=5)
Solution:
If (f(x)=g(x)),then: (ln(f(x))=ln(g(x))→ln(e^x)=ln(5) )
Use log rule: (log_{a}{x^b}=b log_{a}{x}), then: (ln(e^x)=x ln(e)→xln(e)=ln(5) )
(ln(e)=1), then: (x=ln(5) )
Natural Logarithms, Example 4:
Solve equation for (x): (ln(5x-1)=1)
Solution:
Use log rule: (a=log_{b}{b^a}), then: (1=ln(e^1 )=ln(e)→ln(5x-1)=ln(e))
When the logs have the same base: (log_{b}{f(x)}=log_{b}{g(x)}), then: (f(x)=g(x))
then: (ln(5x-1)=ln(e)), then: (5x-1=e→x=frac{e+1}{5})
Exercises to practice Natural Logarithms
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Solve each equation for (x).
- (color{blue}{e^x=3})
- (color{blue}{e^x=4})
- (color{blue}{e^x=8})
- (color{blue}{ln x=6})
- (color{blue}{ln (ln x)=5})
- (color{blue}{e^x=9})
- (color{blue}{ln(2x+5)=4})
- (color{blue}{ln(2x-1)=1})
Answers
- (color{blue}{x=ln 3})
- (color{blue}{x=ln 4,x=2ln(2)})
- (color{blue}{x=ln 8,x=3ln(2)})
- (color{blue}{x=e^6})
- (color{blue}{x=e^{e^5}})
- (color{blue}{x=ln 9,x=2ln(3)})
- (color{blue}{x=frac{e^4-5}{2}})
- (color{blue}{x=frac{e+1}{2}})
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