Final answer:
To solve 2x + 1 = 9, we isolate 2x to get 2x = 8, and then apply ln to both sides, which simplifies to x = ln(8) / ln(2). Using a calculator, we find that x ≈ 3.000.
Step-by-step explanation:
To solve the equation 2x + 1 = 9 using logarithms, first isolate the variable term by subtracting 1 from both sides, resulting in 2x = 8. Then, apply the logarithm to both sides of the equation, where you could use any base for the logarithm. Let's use the natural logarithm (ln), so we have ln(2x) = ln(8). We know that the logarithm of a product can be expressed as the sum of the logarithms (ln xy = ln x + ln y), which gives us x · ln(2) = ln(8). Solving for x, we get x = ln(8) / ln(2). If we had to use the change of base formula, which states logby = log y / log b, our approach would be similar, first expressing 2 as eln(2) and then using properties of exponents to solve for x.
To find the exact value of x, we can use a calculator to compute ln(8) and ln(2), and then divide the former by the latter. If we round our answer to the nearest thousandth, we get x ≈ 3.000, since ln(8) ≈ 2.079 and ln(2) ≈ 0.693, giving us 2.079 / 0.693 ≈ 3.