Change Of Base Formula
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Question 1 of 4
1. Question
Derive the Change of Base formula from the general equation`x=log_a N`Hint
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Well Done!
Incorrect
Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Transform the general logarithmic equation to exponent form$$x$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ $$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ Insert logarithms of the same base to both sides, then solve for `x`$$\color{#D800AD}{N}$$ `=` $$\color{#D800AD}{a^x}$$ $$\log_b \color{#D800AD}{N}$$ `=` $$\log_b \color{#D800AD}{a^x}$$ $$\log_b N$$ `=` $$\log_b a^\color{#004ec4}{x}$$ $$\log_b N$$ `=` $$\color{#004ec4}{x}\log_b a$$ `log_b x^p=p log_b x` `log_b N``divide log_b a` `=` `x log_b a``divide log_b a` Divide both sides by `log_b a` `(log_b N)/(log_b a)` `=` `x` `x` `=` `(log_b N)/(log_b a)` Also, remember that, `x=log_a N`Hence, the Change of Base formula is: `log_a N=(log_b N)/(log_b a)``log_a N=(log_b N)/(log_b a)` -
Question 2 of 4
2. Question
Evaluate using Change of Base`log_2 9`Round answer to `5` decimal places- `log_2 9=` (3.16993)
Hint
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Nice Job!
Incorrect
Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$Change of Base Formula
$$\log_\color{#9a00c7}{a} \color{#00880A}{N}=\frac{\log_b \color{#00880A}{N}}{\log_b \color{#9a00c7}{a}}$$Use the change of base formula, then use the calculator to solve the logarithm$$\log_\color{#9a00c7}{2} \color{#00880a}{9}$$ `=` $$\frac{\log_{10} \color{#00880a}{9}}{\log_{10} \color{#9a00c7}{2}}$$ Calculators use `10` as base for the log function `=` `3.16993` Compute using the calculator `log_2 9=3.16993` -
Question 3 of 4
3. Question
Solve for `x` using Change of Base`5^x=11`Round answer to `4` decimal places- `x=` (1.4899)
Hint
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Excellent!
Incorrect
Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$Change of Base Formula
$$\log_\color{#9a00c7}{a} \color{#00880A}{N}=\frac{\log_b \color{#00880A}{N}}{\log_b \color{#9a00c7}{a}}$$Transform the given exponential equation to logarithmic form$$\color{#9a00c7}{5}^x$$ `=` $$\color{#00880a}{11}$$ $$x$$ `=` $$\log_\color{#9a00c7}{5} \color{#00880a}{11}$$ Use the change of base formula, then use the calculator to solve the logarithm$$x$$ `=` $$\log_\color{#9a00c7}{5} \color{#00880a}{11}$$ $$x$$ `=` $$\frac{\log_{10} \color{#00880a}{11}}{\log_{10} \color{#9a00c7}{5}}$$ Calculators use `10` as base for the log function `x` `=` `1.4899` Compute using the calculator `x=1.4899` -
Question 4 of 4
4. Question
Solve for `x` using Change of Base`2^x=0.062`Round answer to `4` decimal places- `x=` (-4.0116)
Hint
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Fantastic!
Incorrect
Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$Change of Base Formula
$$\log_\color{#9a00c7}{a} \color{#00880A}{N}=\frac{\log_b \color{#00880A}{N}}{\log_b \color{#9a00c7}{a}}$$Transform the given exponential equation to logarithmic form$$\color{#9a00c7}{2}^x$$ `=` $$\color{#00880a}{0.062}$$ $$x$$ `=` $$\log_\color{#9a00c7}{2} \color{#00880a}{0.062}$$ Use the change of base formula, then use the calculator to solve the logarithm$$x$$ `=` $$\log_\color{#9a00c7}{2} \color{#00880a}{0.062}$$ $$x$$ `=` $$\frac{\log_{10} \color{#00880a}{0.062}}{\log_{10} \color{#9a00c7}{2}}$$ Calculators use `10` as base for the log function `x` `=` `-4.0116` Compute using the calculator `x=-4.0116`
Quizzes
- Convert Between Logarithmic and Exponent Form 1
- Convert Between Logarithmic and Exponent Form 2
- Evaluate Logarithms 1
- Evaluate Logarithms 2
- Evaluate Logarithms 3
- Expand Log Expressions
- Simplify Log Expressions 1
- Simplify Log Expressions 2
- Simplify Log Expressions 3
- Logarithmic Equations 1
- Logarithmic Equations 2
- Logarithmic Equations 3
- Change Of Base Formula
- Solving Exponential Equations Using Log Laws