Question

a^{\log _{a} M}= _______

Short answer

M

Step-by-step solution

Understanding the logarithmic expression

Given the expression is \(a^{\log _{a} M}\). Here, \(\log_{a} M\) is the logarithm of \(M\) with base \(a\).

Properties of logarithms and exponents

One important property to remember is that \(a^{\log_{a} x} = x\). This property allows us to simplify expressions involving exponents and logarithms with the same base.

Applying the property

Using the property \(a^{\log_{a} x} = x\), we can simplify the given expression \(a^{\log_{a} M}\). Therefore, \(a^{\log _{a} M} = M\).

Key concepts

logarithms

Logarithms are a fundamental concept in mathematics, often used to solve exponential equations. The logarithm of a number is the exponent to which the base must be raised to produce that number. For instance, in the term \(\log_a M\), \(a\) is the base and \(M\) is the resultant value. This expression reads as 'the logarithm of \(M\) with base \(a\)'. Logarithms are critical because they can transform multiplicative relationships into additive ones, making complex calculations more manageable. They are widely used in fields ranging from science to finance. Understanding the properties and behaviors of logarithms is key to mastering their application.

exponents

Exponents represent repeated multiplication. For example, \(a^3\) means \(a \times a \times a\). In general, \(a^n\) means the base \(a\) is multiplied by itself \(n\) times. This concept is closely tied to logarithms, especially when dealing with equations where a variable is an exponent. When learning about exponents, it's essential to understand their basic properties:

• \(a^0 = 1\) for any non-zero \(a\).
• \(a^1 = a\).
• \(a^{-n} = \frac{1}{a^n} \).

Recognizing these properties helps in simplifying and solving expressions involving exponents. Exponents are everywhere in math and science, from calculating compound interest to understanding geometric transformations.

properties of logarithms

The properties of logarithms are essential tools to simplify and solve logarithmic and exponential problems. Here are some key properties:

• Product Property: \(\log_a (xy) = \log_a x + \log_a y\).
• Quotient Property: \(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\).
• Power Property: \(\log_a (x^n) = n \log_a x\).
• Change of Base Formula: \(\log_b a = \frac{\log_c a}{\log_c b}\).

Moreover, a critical property used in solving the given exercise is \(a^{\log_{a} x} = x\). This property allows us to simplify expressions where the exponent is a logarithmic expression with the same base. By applying these properties, complex logarithmic expressions can turn into simpler ones, making them easier to handle and understand.