Name: power law and change-of-base formula
Uploaded: 2025-11-26T05:43:41.000Z
Duration: 1 min 28 s
Channel: JoVE

Logarithmic laws simplify logarithmic and exponential expressions, especially through the power law.The power law states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of that number.This rule is especially useful when the unknown is in the exponent, such as when solving for time in compound interest, where the amount grows by repeatedly multiplying the principal by a growth factor over time based on the interest rateDividing both sides by the principal isolates the growth factor raised to the time exponent. Taking logarithms on both sides and applying the power law isolates the exponent, allowing time to be solved.Another important property, the change of base formula, is derived using the quotient and power laws. It rewrites a logarithm in one base as a ratio of logarithms in another base.This transformation is useful when a calculator only supports base 10 logarithms (log) and base e logarithms (ln).For example, the change of base formula expresses the logarithm base 4 of 50 as the log base 10 of 50 divided by the log base 10 of 4, allowing computation with standard calculators.

laws of logarithms ii

Laws of Logarithms II

Logarithmic laws provide essential tools for simplifying and evaluating exponential expressions, particularly in mathematical and applied settings where powers and repeated multiplication play a central role. Two important rules are the power law and the change-of-base formula, both allowing for transforming expressions into more manageable forms.The power law of logarithms states that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base number. In symbolic form, this is written as&#160;<figure class='image image_resized image-style-align-center' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41034/41034_Equation_1_editor_41034.svg' alt='41034_Equation_1.svg' /></figure>This rule is especially useful in real-world contexts such as compound interest, where an initial amount grows over time through repeated multiplication. To solve for time, the equation is typically rearranged by dividing both sides by the principal, isolating the growth factor. Taking logarithms of both sides sets up the expression for simplification, and the power law allows the exponent, which often represents time, to be extracted from the exponent position and solved as a linear variable.The change-of-base formula enables the evaluation of logarithms with arbitrary bases by expressing them in terms of logarithms with a base supported by common calculators, such as base 10 or base e. The formula is written as&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41034/41034_Equation_2_editor_41034.svg' alt='41034_Equation_2.svg' /></figure>For example, calculating log&#8289;4(50) becomes&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41034/41034_Equation_3_editor_41034.svg' alt='41034_Equation_3.svg' /></figure>This conversion allows accurate computation using standard logarithmic functions available on most calculators.

Logarithmic laws provide essential tools for simplifying and evaluating exponential expressions, particularly in mathematical and applied settings where ...

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Math Fundamentals

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Exponential and Logarithmic Functions

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power law and change-of-base formula

Logarithmic laws help simplify exponential expressions and make them easier to evaluate. Two key tools are the power law and the change-of-base formula. These rules are useful in math and in applied settings where powers and repeated multiplication appear often.

The power law says that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the number. This rule is helpful when solving equations that involve growth over time, such as compound interest. In that setting, the first step is often to divide by the principal to isolate the growth factor. Then logarithms are taken on both sides, and the power law lets the exponent move out front so time can be solved as a linear variable.

The change-of-base formula helps evaluate logarithms with bases that calculators do not directly support. It rewrites a logarithm in terms of common calculator bases, such as base 10 or base e. For example, log 4 (50) can be converted into a form that uses standard logarithmic functions. This makes it possible to calculate the value accurately with a regular calculator.

Power Law and Change-of-Base Formula

introduction to exponential functions

Exponential Growth and Decay Models

Exponential functions model growth and decay when the rate of change depends on the current value. They are written as f(x) = b^x, where b is a positive constant and not equal to 1. When the base b is greater than 1, the function shows growth. When 0 &lt; b &lt; 1, the function shows decay. These models are useful in many real situations. They can describe bacterial proliferation, radioactive substance decay, and the depreciation of assets. In each case, the change is proportional to the...

exponential functions with base e

Exponential Growth and Decay in Real Life

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introduction to logarithmic functions

Logarithms as Exponents Reversed

Logarithmic functions are the inverse of exponential functions. They are used to solve for exponents. In the general form y = log_a(x), the base a must be greater than 0 and not equal to 1. A logarithm tells you the power needed to get a given number. In other words, y = log_a(x) gives the exponent that the base a must be raised to in order to obtain x. The function is defined only for x &gt; 0, but its range includes all real numbers. The graphs of logarithmic and exponential functions are...

laws of logarithms i

Logarithm Rules for Product and Quotient

Logarithm rules help simplify expressions that use powers, multiplication, and division. A logarithm is the inverse of exponentiation, so it shows how many times a base must be raised to reach a number. For base 10, the common logarithm is written as log. If 10^n = x, then log(x) = n. These rules are useful because they turn harder calculations into simpler steps. The product law says that the logarithm of a product equals the sum of the logarithms. For example, log(100) can be rewritten as...

Logarithmic laws help simplify exponential expressions and make them easier to evaluate. Two key tools are the power law and the change-of-base formula. These rules are useful in math and in applied settings where powers and repeated multiplication appear often. The power law says that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the number. This rule is helpful when solving equations that involve growth over time, such as compound interest.

exponential equations with logarithms: problem solving

exponential-equations-with-logarithms-problem-solving

Solving Population Growth with Logarithms

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applications of logarithms

Logarithms in Sound and Decibels

Logarithms help describe sound intensity with the decibel scale. They are useful when a quantity changes over a very wide range. In acoustics, the decibel (dB) scale turns sound levels into numbers that are easier to compare and use. Sound can range from a faint whisper to the roar of a jet engine. A linear scale would make those differences hard to manage. The decibel scale is a logarithmic scale, which means it measures sound intensity level relative to a reference value. For air, that...

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Exponential growth in population models describes rapid, multiplicative change. It is useful when a population grows by the same factor at regular intervals. This makes it a strong fit for early-stage growth in natural systems. A bacterial culture that doubles every three hours can be modeled with an exponential equation. The model is n(t)=n0·2^(t/3), where n(t) is the population at time t and n0 is the starting population. The base 2 matches the doubling pattern. A more general exponential...

key terms and concepts

Learn how the logarithm power law simplifies exponents and how the change-of-base formula lets you evaluate logs with standard calculator functions.

Power Law and Change-of-Base Formula | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Power Law and Change-of-Base Formula

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