Photo AI

Find, to 3 significant figures, the value of x for which 5^x = 7 - Edexcel - A-Level Maths Pure - Question 6 - 2008 - Paper 2

Question 6

Find, to 3 significant figures, the value of x for which 5^x = 7. Solve the equation 5^2 - 12(5^x) + 35 = 0.

Worked Solution & Example Answer:Find, to 3 significant figures, the value of x for which 5^x = 7 - Edexcel - A-Level Maths Pure - Question 6 - 2008 - Paper 2

Step 1

96%

114 rated

Answer

To solve the equation $5^x = 7$ , we can take the logarithm of both sides. Using logarithm properties:

ext{Taking log:} & \\ x imes ext{log}_b(5) & = ext{log}_b(7) \\ x & = \frac{ ext{log}_b(7)}{ ext{log}_b(5)} \\ x & \approx 1.2091\end{align*}$$ Rounding this value to 3 significant figures, we find: $$x \approx 1.21$$

Step 2

99%

104 rated

Answer

We start with the equation:

$5^2 - 12(5^x) + 35 = 0$ Substituting $5^x = y$ , we rewrite the equation as:

$y^2 - 12y + 35 = 0$

Next, we can factor this quadratic:

$(y - 7)(y - 5) = 0$

This gives us:

$y = 7 \quad \text{or} \quad y = 5$

Substituting back for $5^x$ :

If $5^x = 7$ , then $x \approx \log_5(7) \approx 1.2091 \\ \Rightarrow x \approx 1.21$ (to 3 significant figures)
If $5^x = 5$ , then $x = 1$ .

Thus, the solutions for x are approximately $x \approx 1.21$ and $x = 1$ .

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;