Expand Log Expressions
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Question 1 of 3
1. Question
Expand$$\log_{a}{\frac{xy}{z}}$$Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$Expand the fraction by transforming it into a difference$$\log_{\color{#9a00c7}{a}} \frac{\color{#00880A}{xy}}{\color{#e65021}{z}}$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880A}{xy}- \log_{\color{#9a00c7}{a}} \color{#e65021}{z}$$ Expand the first term by transforming it into a sum$$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{y}- \log_a z$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{y}- \log_a z$$ $$\log_a x + \log_a y- \log_a z$$ -
Question 2 of 3
2. Question
Expand$$\log_a x\sqrt{x+4}$$Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Expand the expression by transforming it into a sum$$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{\sqrt{x+4}}$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{\sqrt{x+4}}$$ Expand further by using the following laws$$\log_b x^\color{#004ec4}{p}$$ `=` $$\color{#004ec4}{p}\log_b x$$ `log_a x+log_a sqrt(x+4)` `=` $$\log_a x+\log_a (x+4)^\color{#CC0000}{\frac{1}{2}}$$ Change the surd into an exponent `=` `log_a x+` `1/2``log_a (x+4)` `log_b x^p=p log_b x` $$\log_a x + \frac{1}{2}\log_a (x+4)$$ -
Question 3 of 3
3. Question
Expand$$\log_{a}{\frac{x(y+z)}{a^3}}$$Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} – \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Expand the fraction by transforming it into a difference$$\log_{\color{#9a00c7}{a}} \frac{\color{#00880A}{x(y+z)}}{\color{#e65021}{a^3}}$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880A}{x(y+z)}-\log_{\color{#9a00c7}{a}} \color{#e65021}{a^3}$$ Expand the first term by transforming it into a sum$$\log_{\color{#9a00c7}{a}} \color{#00880A}{x}\color{#e65021}{(y+z)}- \log_a a^3$$ `=` $$\log_{\color{#9a00c7}{a}} \color{#00880A}{x} + \log_{\color{#9a00c7}{a}} \color{#e65021}{(y+z)}- \log_a a^3$$ Expand further by using the following laws$$\log_b x^\color{#004ec4}{p}$$ `=` $$\color{#004ec4}{p}\log_b x$$ $$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}$$ `=` $$1$$ `log_a x+log_a (y+z)-log_a a^3` `=` `log_a x+log_a (y+z)-` `3``log_a a` `log_b x^p=p log_b x` `=` `log_a x+log_a (y+z)-3(``1``)` `log_b b=1` `=` `log_a x+log_a (y+z)-3` `log_a x+log_a (y+z)-3`
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