[table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Logarithmic Rules[/td][br][td]対数法則[/td][br][td]로그 규칙[/td][br][td]对数规则[/td][br][/tr][br][tr][br][td]Base Change Rule[/td][br][td]底の変換規則[/td][br][td]밑변환 법칙[/td][br][td]底数变换规则[/td][br][/tr][br][tr][br][td]Simplifying Expressions[/td][br][td]式の簡素化[/td][br][td]표현식 단순화[/td][br][td]简化表达式[/td][br][/tr][br][tr][br][td]Exponential Equations[/td][br][td]指数方程式[/td][br][td]지수 방정식[/td][br][td]指数方程[/td][br][/tr][br][tr][br][td]Logarithmic Equations[/td][br][td]対数方程式[/td][br][td]로그 방정식[/td][br][td]对数方程[/td][br][/tr][br][tr][br][td]Indices Rules[/td][br][td]指数法則[/td][br][td]지수 규칙[/td][br][td]指数规则[/td][br][/tr][br][tr][br][td]Natural Logarithm, ln[/td][br][td]自然対数[/td][br][td]자연로그[/td][br][td]自然对数[/td][br][/tr][br][/table][br]

[table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What are the basic logarithmic rules, including the product, quotient, and power rules?[br] [*]How can logarithmic rules be used to simplify expressions involving logarithms?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]Why do logarithmic rules work, and how do they relate to the properties of exponents?[br] [*]How can understanding logarithmic rules aid in solving exponential and logarithmic equations?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Is the emphasis on learning logarithmic rules in high school mathematics justified by their application in higher mathematics and real-world problems?[br] [*]Can the principles behind logarithmic rules be extended to simplify complex problems in fields such as information theory and computational complexity?[br] [/list][br] [/td][br][/tr][br][/table][br]

Mini-Investigation: Discovering Logarithmic Rules[br][br]Objective: [br][br]Use the interactive applet provided to explore and uncover the fundamental rules of logarithms. This investigation will guide you through a series of steps to understand how logarithms work and how their properties are applied.[br][br]Investigation Steps:[br][br]Step 1: Discovering the Product Rule[br]- Use the applet to calculate the log of two numbers multiplied together (e.g., [math]log(2\times3)[/math]).[br]- Next, calculate the log of each number separately (log(2) and log(3)) and then add these values together.[br]- Compare the results. What do you notice? This observation will lead you to the Product Rule of logarithms.[br][br]Step 2: Investigating the Quotient Rule[br]- Repeat a similar process as in Step 2, but this time use division. Calculate the log of a number divided by another (e.g., [math]log(\frac{8}{2})[/math]).[br]- Calculate the log of the numerator and the denominator separately ([math]log(8)[/math] and [math]log(2)[/math]) and then subtract the denominator's log from the numerator's log.[br]- Observe and note the relationship between these values, leading you to the Quotient Rule of logarithms.[br][br]Step 3: Uncovering the Power Rule[br]- Use the applet to calculate the log of a number raised to a power (e.g., [math]log(2^3)[/math]).[br]- Calculate the log of the base number ([math]log(2)[/math]) and then multiply it by the exponent ([math]3[/math]).[br]- Examine how the calculated log compares to the log of the number raised to a power. This will help you understand the Power Rule of logarithms.[br][br]Step 4: Confirming the Base Change Rule[br]- Experiment with changing the base of the logarithm in the applet.[br]- Try to express a logarithm of one base in terms of logarithms of another base using the applet's outputs. [br][br]Conclusion:[br]- Summarize the rules you discovered: Product Rule, Quotient Rule, Power Rule, and the Base Change Rule.[br]- Reflect on how these rules help in simplifying complex logarithmic expressions and in solving logarithmic equations.[br][br]

Checking understanding of simplifying log expressions

Alternatively these more bite-size videos give worked examples of test style questions. [br][br]Indices rules - This is basic but incredibly important skill that is used later in calculus  [url=https://youtu.be/TOXhuEoEi44]https://youtu.be/TOXhuEoEi44[/url], Exponentials - Exponential equations (hidden quadratic) , [url=https://youtu.be/12BD5yk4zso]https://youtu.be/12BD5yk4zso[/url] , Logarithms  [url=https://youtu.be/C2f5NvtMCoU]https://youtu.be/C2f5NvtMCoU[/url] , Natural log - ln(x) , [url=https://youtu.be/dl7mOHtOKu4]https://youtu.be/dl7mOHtOKu4[/url] Using logs to find unknown powers , [url=https://youtu.be/7sBzT9XdDdo]https://youtu.be/7sBzT9XdDdo[/url]  

Testing your understanding using exam-style questions