Example 7.1.2: Solving Exponential Equations Using Logarithms | 示例7.1.2：利用对数求解指数方程

📚 Example 7.1.2: Solving Exponential Equations Using Logarithms | 示例7.1.2：利用对数求解指数方程

In this worked example, we will solve the equation 2ˣ = 5. This equation is exponential because the unknown x appears in the exponent. Solving such equations is a core skill in A-Level Mathematics. The method involves applying logarithms, which are the inverse operations of exponentiation. We will explain each stage carefully, from taking logs to verifying the solution.

在这个例题中，我们将求解方程 2ˣ = 5。该方程是指数方程，因为未知数 x 出现在指数位置上。求解这类方程是 A-Level 数学的核心技能。方法涉及使用对数，对数是指数运算的逆运算。我们将逐步详细解释，从取对数到验证解。

1. Understanding the Equation | 理解方程

The equation 2ˣ = 5 states that 2 raised to the power x equals 5. The base is 2, the exponent is x, and the value of the expression is 5. We need to determine the exact value of x.

方程 2ˣ = 5 表示 2 的 x 次幂等于 5。底数是 2，指数是 x，表达式的值是 5。我们需要确定 x 的具体值。

Because x is not a simple coefficient, we cannot solve it by basic algebraic rearrangements. The variable is trapped in the exponent, so we need a tool that can “bring it down” to ground level. That tool is a logarithm.

因为 x 不是简单的系数，我们无法通过基本的代数变形来求解。变量被限制在指数中，所以我们需要一种工具将其”降”到普通位置。这个工具就是对数。

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2. Why Use Logarithms? | 为什么使用对数？

A logarithm answers the question: “To what exponent must a given base be raised to obtain a certain number?” For example, log₂ 8 = 3 because 2³ = 8. This property makes logs ideal for undoing exponentials.

对数回答的问题是：”一个给定的底数需要被提升到多少次幂，才能得到某个数？” 例如，log₂ 8 = 3，因为 2³ = 8。这一性质使得对数非常适合用来解除指数。

In our equation, the base is 2 and the result is 5. If we take the logarithm of both sides with any consistent base, the exponent x will move down as a multiplier. We usually use base 10 or base e (natural logarithm) because these are available on scientific calculators.

在我们的方程中，底数是 2，结果是 5。如果我们对方程两边取任意相同底数的对数，指数 x 就会作为乘数被移下来。我们通常使用底数 10 或底数 e（自然对数），因为这些在科学计算器上可以直接使用。

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3. Taking Logarithms of Both Sides | 两边取对数

We begin by taking the logarithm of both sides of the equation. Using base 10, we write:

我们首先对方程两边取对数。使用底数 10，我们写：

log₁₀(2ˣ) = log₁₀(5)

This step is valid because logarithm is a function: if two quantities are equal, their logarithms to the same base are also equal.

这一步是有效的，因为对数是一个函数：如果两个量相等，那么它们相对于同一底数的对数也相等。

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4. Applying the Power Rule | 应用幂规则

The power rule of logarithms states that logₐ(mⁿ) = n logₐ m. Applying this to the left-hand side gives:

对数的幂规则指出：logₐ(mⁿ) = n logₐ m。将此法则应用于左边，得到：

x log₁₀ 2 = log₁₀ 5

Notice how x is no longer in the exponent. It has been rewritten as a coefficient, which allows us to isolate it using ordinary algebra.

注意 x 不再位于指数位置。它被改写为系数，这样我们就可以使用普通代数将其分离出来。

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5. Isolating x | 解出 x

Now we divide both sides of the equation by log₁₀ 2 (which is a non-zero constant) to obtain:

现在我们将方程两边除以 log₁₀ 2（这是一个非零常数），得到：

x = log₁₀ 5 / log₁₀ 2

This is the exact solution. No further simplification is needed unless we wish to evaluate the logarithms.

这就是精确解。除非我们想计算对数值，否则无需进一步化简。

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6. Using a Calculator | 使用计算器

To obtain a decimal approximation, use a scientific calculator. Enter log₁₀ 5, then divide by log₁₀ 2. The result is approximately 2.3219280949.

为了得到十进制近似值，使用科学计算器。输入 log₁₀ 5，然后除以 log₁₀ 2。结果约为 2.3219280949。

x ≈ 2.3219 (to 4 decimal places)

If your calculator has a natural logarithm button (ln), you could also use ln 5 / ln 2. The ratio is the same because a change of base does not affect the result.

如果你的计算器有自然对数按钮（ln），你也可以使用 ln 5 / ln

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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