Understanding Logarithms and Exponents | Solving Equations with Logarithms and Exponents

solving equations with logarithms and exponents

To solve equations involving logarithms and exponents, we need to understand the properties and rules of logarithms and exponents

To solve equations involving logarithms and exponents, we need to understand the properties and rules of logarithms and exponents.

Logarithms:
A logarithm is the inverse operation of raising a number to a certain power. It tells us what exponent we need to raise a given base to in order to obtain a certain number.
The basic logarithmic equation is:
log(base a) x = y

In this equation, “a” is the base, “x” is the number we want to find the logarithm of, and “y” is the exponent or power to which we raise the base “a” to get “x”.

Properties of Logarithms:
1. Product Rule: log(base a) (x * y) = log(base a) x + log(base a) y
2. Quotient Rule: log(base a) (x / y) = log(base a) x – log(base a) y
3. Power Rule: log(base a) x^y = y * log(base a) x
4. Change of Base Rule: log(base a) x = log(base b) x / log(base b) a

Exponents:
An exponent represents how many times a number is to be multiplied by itself. It is written as a small number raised to the right and above the base number.
The basic exponential equation is:
a^x = y

In this equation, “x” is the exponent, “a” is the base, and “y” is the result of raising the base “a” to the power of “x”.

Now let’s solve an equation involving logarithms and exponents as an example:

Example:
Solve for x: log(base 2) (3x + 4) = 5

Step 1: Rewrite the equation in exponential form using the definition of logarithms.
2^5 = 3x + 4

Step 2: Simplify the exponential equation.
32 = 3x + 4

Step 3: Isolate the variable term “3x”.
32 – 4 = 3x
28 = 3x

Step 4: Solve for x by dividing both sides of the equation by 3.
28/3 = x
x = 9.333 (rounded to three decimal places)

Thus, the solution of the equation log(base 2) (3x + 4) = 5 is x = 9.333.

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