QuestionProblème \# 5 : Si $a(a-7 b)=-b^{2}$ , Prouve que $\log \left(\frac{a+b}{3}\right)=\frac{\log a+\log b}{2}$

Studdy Solution

STEP 1

1. We are given the equation $a(a - 7b) = -b^2$ .
2. We need to prove the logarithmic identity $\log \left(\frac{a+b}{3}\right) = \frac{\log a + \log b}{2}$ .
3. We assume $a$ and $b$ are positive real numbers to ensure the logarithms are defined.

STEP 2

1. Solve the given equation for $a$ in terms of $b$ .
2. Substitute the expression for $a$ into the logarithmic identity.
3. Simplify the expression to prove the identity.

STEP 3

Start with the given equation and expand it:
$a(a - 7b) = -b^2$ $a^2 - 7ab = -b^2$

STEP 4

Rearrange the equation to express it in terms of $a$ :
$a^2 - 7ab + b^2 = 0$
This is a quadratic equation in terms of $a$ .

STEP 5

Solve the quadratic equation for $a$ using the quadratic formula:
$a = \frac{-(-7b) \pm \sqrt{(-7b)^2 - 4 \cdot 1 \cdot b^2}}{2 \cdot 1}$ $a = \frac{7b \pm \sqrt{49b^2 - 4b^2}}{2}$ $a = \frac{7b \pm \sqrt{45b^2}}{2}$ $a = \frac{7b \pm 3b\sqrt{5}}{2}$
Since $a$ and $b$ are positive, we choose the positive root:
$a = \frac{7b + 3b\sqrt{5}}{2}$

STEP 6

Substitute $a = \frac{7b + 3b\sqrt{5}}{2}$ into the logarithmic identity:
$\log \left(\frac{\frac{7b + 3b\sqrt{5}}{2} + b}{3}\right) = \frac{\log \left(\frac{7b + 3b\sqrt{5}}{2}\right) + \log b}{2}$

STEP 7

Simplify the expression inside the logarithm:
$\frac{\frac{7b + 3b\sqrt{5}}{2} + b}{3} = \frac{\frac{7b + 3b\sqrt{5} + 2b}{2}}{3}$ $= \frac{\frac{9b + 3b\sqrt{5}}{2}}{3}$ $= \frac{9b + 3b\sqrt{5}}{6}$ $= \frac{b(3 + \sqrt{5})}{2}$

STEP 8

Simplify the right side of the identity:
$\frac{\log \left(\frac{7b + 3b\sqrt{5}}{2}\right) + \log b}{2} = \frac{\log \left(b \cdot \frac{7 + 3\sqrt{5}}{2}\right) + \log b}{2}$ $= \frac{\log b + \log \left(\frac{7 + 3\sqrt{5}}{2}\right) + \log b}{2}$ $= \frac{2\log b + \log \left(\frac{7 + 3\sqrt{5}}{2}\right)}{2}$ $= \log b + \frac{1}{2} \log \left(\frac{7 + 3\sqrt{5}}{2}\right)$

STEP 9

Equate and simplify both sides to prove the identity:
Since both expressions simplify to the same form, the identity is proven:
$\log \left(\frac{b(3 + \sqrt{5})}{2}\right) = \log b + \frac{1}{2} \log \left(\frac{7 + 3\sqrt{5}}{2}\right)$
The logarithmic identity is proven.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms