Solving equations Solve the following equations.

7ˣ = 21

검증된 단계별 안내

Step 1: Recognize that the equation \(7^x = 21\) involves an exponential function, and to solve for \(x\), we need to use logarithms.

Step 2: Take the natural logarithm (or common logarithm) of both sides of the equation: \(\ln(7^x) = \ln(21)\).

Step 3: Apply the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\) to the left side of the equation: \(x \cdot \ln(7) = \ln(21)\).

Step 4: Solve for \(x\) by dividing both sides of the equation by \(\ln(7)\): \(x = \frac{\ln(21)}{\ln(7)}\).

Step 5: The expression \(x = \frac{\ln(21)}{\ln(7)}\) gives the solution for \(x\) in terms of logarithms.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In the equation 7ˣ = 21, we are dealing with an exponential function where the variable 'x' is in the exponent. Understanding the properties of exponential functions is crucial for solving equations involving them.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations like 7ˣ = 21. The logarithm log_b(a) answers the question: to what exponent must the base 'b' be raised to produce 'a'? In this case, we can use logarithms to isolate 'x' by rewriting the equation in logarithmic form.

Properties of Equality

The properties of equality state that if two expressions are equal, then we can perform the same operation on both sides without changing the equality. This principle is essential when manipulating equations, such as applying logarithms to both sides of 7ˣ = 21 to solve for 'x'. Understanding these properties ensures that the solutions derived from the equations are valid.

Solving equations Solve the following equations.7ˣ = 21