Study guides › 0606 › Logarithmic and exponential functions Maths4U Original · Learn, check, explain
General logarithms and laws — 0606
Translate a logarithm into an exponent and combine positive arguments.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods For \(\displaystyle a>0, a\ne 1\) and x>0, \(\log_a x=y\) means \(\displaystyle a^{y}=x\). The product, quotient and power laws are \(\log_a(uv)=\log_a u+\log_a v\), \(\displaystyle \log_{a} (\frac{u}{v})= \log_{a} u- \log_{a} v\), and \(\log_a(u^r)=r\log_a u\) when the real expressions are defined.
Natural logarithms use base e. Exponential and logarithmic graphs are inverse reflections in \(\displaystyle y=x\). The logarithm argument must remain positive; combining expressions does not remove that original restriction.
Check the conditions Logarithm arguments are positive. The base is positive and different from one. Three worked examples Example 1 · Logarithms and index laws Let \(\displaystyle a=\)\(2\), \(p = 64\), \(q = 16\). Work exactly, using index laws.
Find \(\displaystyle \log _{a} p\). Find logₐ(pq). Find \(\displaystyle \log_{a} (\frac{p}{q})\). Find \(\displaystyle \log_{a} (\frac{p^{2}}{q})\). Show the reasoning (a) Apply the product, quotient or power law to positive arguments.
6
Answer: \(6\).
(b) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 6+4=10\)
Answer: \(10\).
(c) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 6-4=2\)
Answer: \(2\).
(d) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 2(6)-4=8\)
Answer: \(8\).
Example 2 · Logarithms and index laws Let \(\displaystyle a=\)\(3\), \(p = 2187\), \(q = 3\). Work exactly, using index laws.
Find \(\displaystyle \log _{a} p\). Find logₐ(pq). Find \(\displaystyle \log_{a} (\frac{p}{q})\). Find \(\displaystyle \log_{a} (\frac{p^{2}}{q})\). Show the reasoning (a) Apply the product, quotient or power law to positive arguments.
7
Answer: \(7\).
(b) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 7+1=8\)
Answer: \(8\).
(c) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 7-1=6\)
Answer: \(6\).
(d) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 2(7)-1=13\)
Answer: \(13\).
Example 3 · Logarithms and index laws Let \(\displaystyle a=\)\(4\), \(p = 64\), \(q = 16\). Work exactly, using index laws.
Find \(\displaystyle \log _{a} p\). Find logₐ(pq). Find \(\displaystyle \log_{a} (\frac{p}{q})\). Find \(\displaystyle \log_{a} (\frac{p^{2}}{q})\). Show the reasoning (a) Apply the product, quotient or power law to positive arguments.
3
Answer: \(3\).
(b) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 3+2=5\)
Answer: \(5\).
(c) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 3-2=1\)
Answer: \(1\).
(d) Apply the product, quotient or power law to positive arguments.
\(\displaystyle 2(3)-2=4\)
Answer: \(4\).
Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
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