A-Level Edexcel Maths Pure Geometric Sequences & Series: 6. (a) Find, to 3 significant fi

6. (a) Find, to 3 significant figures, the value of x for which 8^x = 0.8 - Edexcel - A-Level Maths Pure - Question 8 - 2007 - Paper 2

Question 8

6. (a) Find, to 3 significant figures, the value of x for which 8^x = 0.8. (b) Solve the equation 2 log_1 x - log_3 7x = 1.

Worked Solution & Example Answer:6. (a) Find, to 3 significant figures, the value of x for which 8^x = 0.8 - Edexcel - A-Level Maths Pure - Question 8 - 2007 - Paper 2

Step 1

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Answer

To solve for x in the equation $8^x = 0.8$ , we first take the logarithm on both sides. Using logarithmic properties, we can rewrite this as:

$x \log(8) = \log(0.8)$

Next, we isolate x:

$x = \frac{\log(0.8)}{\log(8)}$

Calculating this, we find:

Now, substituting these values in:

$x \approx \frac{-0.09691}{0.90309} \approx -0.107$

Thus, to 3 significant figures, the value of x is:

Answer: $x = -0.107$ .

Step 2

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Answer

First, we reorganize the equation:

$2 \log_1 x - \log_3 (7x) = 1$

Rewriting the left-hand side, we have:

$2 \log_1 x = 1 + \log_3 (7x)$

Now we can apply the properties of logs to isolate x.

Rearranging, we convert the logarithm with base 3:

$\log_3 (7x) = \log_3 7 + \log_3 x$

This gives:

$2 \log_1 x = 1 + \log_3 7 + \log_3 x$

To simplify, let's set: $\log_1 x = A$ , Thus the equation becomes: $2A = 1 + \log_3 7 + A$

From which we find: $A = 1 + \log_3 7$

Now substituting back: $\log_1 x = 1 + \log_3 7$

To find x, we solve: $x = 21$

Thus, the solution of the equation is:

Answer: $x = 21$ .