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Solving Logarithmic Equations

The Organic Chemistry Tutor・17 minutes read

The text illustrates how to convert various logarithmic equations into exponential form to determine the value of x in each case, demonstrating different techniques for solving them. Key examples include finding x as 4, 3, 125, 16, and other values through appropriate transformations and calculations.

Insights

  • To solve logarithmic equations, it is essential to convert them into exponential form, as demonstrated by multiple examples in the text. For instance, converting log base 2 of 16 to exponential form reveals that x equals 4, while log base x of 81 simplifies to show that x equals 3. This method highlights the fundamental relationship between logarithms and exponents, making it easier to find solutions.
  • Additionally, the text illustrates the diversity of logarithmic equations, including those with different bases and forms, such as natural logarithms and logarithmic identities. For example, solving ln(3x 2) = 5 involves converting to exponential form and leads to a solution of approximately 50.14, while the equation log x^2 = 2 log x demonstrates how factoring can yield multiple solutions. This variety emphasizes the importance of understanding the properties of logarithms and their applications in various mathematical contexts.

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Recent questions

  • What is a logarithm?

    A logarithm is a mathematical function that determines the exponent or power to which a base must be raised to produce a given number. For example, in the expression log base b of a, the logarithm answers the question: "To what power must b be raised to yield a?" This concept is fundamental in various fields, including mathematics, science, and engineering, as it helps simplify complex calculations, particularly those involving exponential growth or decay. Logarithms can be expressed in different bases, such as base 10 (common logarithm) or base e (natural logarithm), and they have unique properties that make them useful for solving equations and analyzing data.

  • How do you solve logarithmic equations?

    Solving logarithmic equations typically involves converting the logarithmic expression into its exponential form. This process allows you to isolate the variable and solve for its value. For instance, if you have an equation like log base b of a = x, you can rewrite it as b^x = a. From there, you can manipulate the equation using algebraic techniques to find the value of x. Additionally, understanding properties of logarithms, such as the product, quotient, and power rules, can aid in simplifying and solving more complex logarithmic equations effectively.

  • What is the change of base formula?

    The change of base formula is a mathematical tool that allows you to convert logarithms from one base to another, making it easier to compute logarithmic values using a calculator that may only support specific bases. The formula states that log base b of a can be expressed as log base c of a divided by log base c of b, where c is any positive number different from 1. This is particularly useful when dealing with logarithms in different bases, as it enables you to calculate logarithmic values in a consistent manner, facilitating comparisons and computations across various logarithmic expressions.

  • What is the natural logarithm?

    The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. This logarithm is significant in mathematics, particularly in calculus and exponential growth models, as it arises naturally in various contexts, such as compound interest and population growth. The natural logarithm has unique properties, including the fact that the derivative of ln(x) is 1/x, making it a fundamental function in calculus. It is widely used in scientific applications, finance, and engineering due to its relationship with exponential functions and its ability to simplify complex calculations.

  • What are logarithmic properties?

    Logarithmic properties are rules that govern the manipulation and simplification of logarithmic expressions. These properties include the product rule, which states that log base b of (xy) equals log base b of x plus log base b of y; the quotient rule, which states that log base b of (x/y) equals log base b of x minus log base b of y; and the power rule, which states that log base b of (x^k) equals k times log base b of x. Understanding these properties is essential for solving logarithmic equations, simplifying expressions, and performing calculations involving logarithms, as they provide a systematic approach to working with these mathematical functions.

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Summary

00:00

Solving Logarithmic Equations Step by Step

  • To solve the logarithmic equation log base 2 of 16 = x, convert it to exponential form: 2^x = 16. Since 2^4 = 16, the value of x is 4. Alternatively, using the change of base formula, log(16)/log(2) also equals 4.
  • For the equation log base x of 81 = 4, convert to exponential form: x^4 = 81. Taking the fourth root of both sides gives x = 3, as 3^4 = 81.
  • In the equation log base 5 of x = 3, convert to exponential form: 5^3 = x. Calculating gives x = 125, since 5^3 = 125.
  • For log base 32 of x = 4/5, convert to exponential form: 32^(4/5) = x. First, find the fifth root of 32, which is 2, then raise it to the fourth power: 2^4 = 16, so x = 16.
  • In the equation log base 3 of (5x + 1) = 4, convert to exponential form: 3^4 = 5x + 1. This simplifies to 81 = 5x + 1. Subtracting 1 gives 80, and dividing by 5 results in x = 16.
  • If log x = 24, the base is assumed to be 10, so x = 10^24, a very large number.
  • For ln x = 7, the base is e, so x = e^7, which is approximately 1096.65.
  • In the equation log base 7 of (x^2 + 3x + 9) = 2, convert to exponential form: 7^2 = x^2 + 3x + 9. This gives 49 = x^2 + 3x + 9. Rearranging leads to x^2 + 3x - 40 = 0, which factors to (x + 8)(x - 5) = 0, yielding x = -8 or x = 5.
  • For the equation ln(3x - 2) = 5, convert to exponential form: e^5 = 3x - 2. Adding 2 and dividing by 3 gives x = (e^5 + 2)/3, approximately 50.14.
  • In the equation 4ln(2x - 1) + 3 = 11, subtract 3 to get 8, then divide by 4 to find 2. Converting to exponential form gives e^2 = 2x - 1. Solving for x results in x = (e^2 + 1)/2, approximately 4.1945.

20:15

Logarithmic Equations and Their Solutions

  • The equation starts with \( \log x^2 = 2 \log x \), which is rearranged to \( 0 = \log x^2 - 2 \log x \). Factoring gives \( \log x (\log x - 2) = 0 \), leading to two solutions: \( \log x = 0 \) (thus \( x = 1 \)) and \( \log x = 2 \) (thus \( x = 100 \)). Both solutions are verified by substituting back into the original equation, confirming \( x = 1 \) and \( x = 100 \) are correct.
  • For the equation \( \log(\log x) = 4 \), it is solved by recognizing the base is 10, leading to \( 10^4 = \log x \) or \( x = 10^{10,000} \). Another example involves \( \log_3(\log_2 x) = 2 \), which simplifies to \( \log_2 x = 9 \) (since \( 3^2 = 9 \)), resulting in \( x = 2^9 = 512 \).
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