Writing Logarithmic Equations In Exponential Form

The Organic Chemistry Tutor
30 Jan 201803:06
EducationalLearning
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TLDRThis lesson teaches the conversion between logarithmic expressions and exponential equations. It begins with an example of converting 'log base 3 of 9 equals 2' to its exponential form, '3^2 = 9'. The video then challenges viewers to convert '4^3 = 64' and '8^2 = 64' into logarithmic form. The method involves identifying the base and the exponent, then setting up the logarithmic equation with the base, the result of the exponentiation, and the exponent itself. The lesson concludes with examples of converting exponential equations like '2^3 = 8', '4^2 = 16', and '5^3 = 125' into their respective logarithmic forms, emphasizing the relationship between the base, exponent, and the result.

Takeaways
  • 📚 The lesson focuses on converting logarithmic expressions into exponential equations.
  • 🔍 A logarithmic expression 'log base a of b is equal to c' can be converted to its exponential form 'a raised to the power of c equals b'.
  • 📐 The example given is converting 'log base 3 of 9 is equal to 2' to '3 squared is equal to 9'.
  • 🧩 The process involves rewriting the logarithmic form to show the base raised to the power of the logarithm result equals the original number.
  • 📈 The script also covers converting exponential equations into logarithmic form.
  • 🔢 For converting exponential to logarithmic, the base of the exponential becomes the base of the logarithm, and the exponent becomes the result of the logarithm.
  • 📝 The example '2 raised to the third power is equal to 8' is converted to 'log base 2 of 8 is equal to 3'.
  • 📉 Another example is 'four raised to the second power is equal to sixteen', which converts to 'log base 4 of 16 is equal to 2'.
  • 📚 The final example is 'five to the third power is equal to 125', which translates to 'log base 5 of 125 is equal to 3'.
  • 🔎 The script encourages the viewer to practice by pausing the video and attempting the conversions themselves.
  • 📋 The lesson provides a clear methodology for converting between logarithmic and exponential forms, emphasizing the relationship between the base, exponent, and the result.
Q & A

  • What is the main focus of the lesson?

    -The main focus of the lesson is converting logarithmic expressions into exponential equations.

  • How can the logarithmic expression 'log base 3 of 9 is equal to 2' be converted into exponential form?

    -The logarithmic expression 'log base 3 of 9 is equal to 2' can be converted into exponential form as '3 raised to the power of 2 equals 9' or '3^2 = 9'.

  • What is the generic equation for converting a logarithmic expression into exponential form?

    -The generic equation for converting a logarithmic expression 'log base a of b is equal to c' into exponential form is 'a raised to the power of c equals b' or 'a^c = b'.

  • What is the exponential form of '4 raised to the third power equals 64'?

    -The exponential form of '4 raised to the third power equals 64' is '4^3 = 64'.

  • What is the exponential form of 'eight raised to the second power equals 64'?

    -The exponential form of 'eight raised to the second power equals 64' is '8^2 = 64'.

  • How can an exponential equation be converted into a logarithmic equation?

    -An exponential equation can be converted into a logarithmic equation by placing the base on the left side of the equation and the number on the right side of the equation with the exponent as the value of the logarithmic function.

  • What is the logarithmic form of the exponential equation '2 raised to the third power equals 8'?

    -The logarithmic form of the exponential equation '2 raised to the third power equals 8' is 'log base 2 of 8 equals 3' or 'log2(8) = 3'.

  • What is the logarithmic form of 'four raised to the second power equals sixteen'?

    -The logarithmic form of 'four raised to the second power equals sixteen' is 'log base 4 of 16 equals 2' or 'log4(16) = 2'.

  • What is the logarithmic form of 'five raised to the third power equals 125'?

    -The logarithmic form of 'five raised to the third power equals 125' is 'log base 5 of 125 equals 3' or 'log5(125) = 3'.

  • How can you check if the conversion from exponential to logarithmic form is correct?

    -You can check if the conversion from exponential to logarithmic form is correct by reversing the process and verifying that the original exponential equation holds true.

Outlines
00:00
📚 Converting Logarithmic to Exponential Form

This paragraph introduces the concept of converting logarithmic expressions into exponential equations. It begins with an example of converting log base 3 of 9 equals 2 into the exponential form 3^2 = 9. The generic formula for conversion is explained as a^c = b, where 'a' is the base, 'b' is the result, and 'c' is the exponent. The paragraph then challenges the viewer to convert 4^3 = 64 and 8^2 = 64 into their respective exponential forms. The process is further elaborated with examples to demonstrate how to convert exponential equations back into logarithmic form, using the formula log base a of b equals c, where 'a' is the base, 'b' is the result, and 'c' is the exponent. The examples provided include converting 2^3 = 8, 4^2 = 16, and 5^3 = 125 into their logarithmic equivalents.

Mindmap
Keywords
💡Logarithmic expression
A logarithmic expression is a mathematical notation that represents the power to which a base number must be raised to produce a given value. In the context of the video, converting a logarithmic expression to an exponential form is the main focus. For example, the script mentions 'log base 3 of 9 is equal to 2', which is a logarithmic expression indicating that 3 raised to the power of 2 equals 9.
💡Exponential form
Exponential form is a way of expressing a mathematical relationship where a number, known as the base, is raised to a power to produce a result. It is the inverse of a logarithmic expression. The video script explains how to convert logarithmic expressions into exponential form, such as 'three raised to the second power is equal to nine', which is the exponential form of the logarithmic expression 'log base 3 of 9 is equal to 2'.
💡Log base
The term 'log base' refers to the base number in a logarithmic expression. It is the number that is raised to a certain power to yield a specific value. In the video script, the base is used to demonstrate the conversion process, such as 'log base 3' where 3 is the base number in the logarithmic expression 'log base 3 of 9 is equal to 2'.
💡Raised to the power
To be 'raised to the power' means to multiply a number by itself a certain number of times, which is the exponent. This concept is central to understanding both logarithmic and exponential forms. In the script, it is used to explain how to convert 'log base 3 of 9' into 'three raised to the second power'.
💡Generic equation
A generic equation is a formula that can be applied to various specific cases. In the video, the generic equation 'a raised to the c is equal to b' is used to demonstrate the conversion from a logarithmic expression 'log base a of b is equal to c' to its exponential form. It serves as a template for understanding the relationship between logarithms and exponents.
💡Conversion
Conversion in the context of the video refers to the process of changing a logarithmic expression into its equivalent exponential form, or vice versa. The script provides examples of converting 'log base 3 of 9 is equal to 2' into 'three squared is equal to nine', illustrating the conversion process.
💡Exponent
An exponent is the power to which a base number is raised in an exponential expression. It is a fundamental concept in the video, as it represents the 'c' in the logarithmic expression 'log base a of b is equal to c'. The script uses exponents to show the relationship between logarithmic and exponential forms, such as in 'three raised to the second power' where 'second' is the exponent.
💡Base
In both logarithmic and exponential expressions, the 'base' is the number that is raised to an exponent. The base is crucial in determining the result of the expression. The video script uses the base to explain how to convert between logarithmic and exponential forms, as seen in 'log base 2' where 2 is the base.
💡Equation
An equation is a mathematical statement that asserts the equality of two expressions. In the video, equations are used to demonstrate the relationship between logarithmic and exponential forms. For example, 'three squared is equal to nine' is an equation that shows the result of converting a logarithmic expression to an exponential form.
💡Check
To 'check' in the context of the video means to verify the correctness of the conversion from a logarithmic expression to an exponential form, or vice versa. The script encourages viewers to check their work by ensuring that the base raised to the exponent equals the original value, as shown in '2 raised to the third power is equal to 8'.
Highlights

Lesson focuses on converting logarithmic expressions into exponential equations.

Logarithmic expression log base 3 of 9 equals 2 is used as an example.

Generic equation for conversion: a raised to the power of c equals b.

Conversion example: 3 squared equals nine.

Instructions to convert 4 raised to the third power equals 64 into logarithmic form.

Conversion result: eight raised to the second power equals 64.

Explanation of converting exponential form to logarithmic form.

Conversion example: log base 2 of 8 equals 3.

Verification of conversion: 2 raised to the third power equals 8.

Conversion example: log base 4 of 16 equals 2.

Verification of conversion: 4 squared equals 16.

Conversion example: log base 5 of 125 equals 3.

Verification of conversion: 5 cubed equals 125.

Encouragement to pause the video and try the examples.

The base in a logarithmic function is the one with the exponent.

The logarithmic function always equals the exponent.

The importance of checking the conversion through verification.

Transcripts

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