Question

Log2(3)=a log2(5)=blog4(135)=?​

Answer 1

$\log_{2}3 = a, \ \log_{2}5 = b$

$\log_{4}135 = \log_{2^{2}}(27 \cdot 5) = \dfrac{1}{2} \log_{2}3^{3} + \dfrac{1}{2}\log_{2}5 =\dfrac{3}{2} \log_{2}3 + \dfrac{1}{2}\log_{2}5 = \dfrac{3}{2} a + \dfrac{1}{2} b$

Свойства логарифмов:

$\log_{a^{p}}b = \dfrac{1}{p}\log_{a}b \ \ \ (a > 0, \ a \neq 1, \ b > 0)$

$\log_{a}b^{p} = p\log_{a}b \ \ \ (a > 0, \ a \neq 1, \ b > 0)$

$\log_{a}(xy) = \log_{a}x + \log_{a}y \ (a>0, \ a\neq 1, \ x > 0, \ y > 0)$

Answer 2

Let's solve these logarithmic equations step by step:

1. Log2(3) = a: This equation is asking for the value of 'a' such that log base 2 of 3 equals 'a'.

   We can write this equation as: 2^a = 3 Taking the logarithm base 2 of both sides: log2(2^a) = log2(3)

   Since log2(2^a) simplifies to 'a', the equation becomes: a = log2(3)

2. Log2(5) = b: This equation is asking for the value of 'b' such that log base 2 of 5 equals 'b'.

   We can write this equation as: 2^b = 5 Taking the logarithm base 2 of both sides: log2(2^b) = log2(5)

   Again, simplifying the left side: b = log2(5)

3. Log4(135) = ?​: This equation is asking for the logarithm base 4 of 135.

   We can write this equation as: 4^x = 135 Taking the logarithm base 4 of both sides: log4(4^x) = log4(135)

   Simplifying the left side: x = log4(135)

Now we have expressions for 'a' and 'b' in terms of logarithms:

a = log2(3) b = log2(5)

And we need to find 'x' for the third equation:

x = log4(135)

If you want a numeric approximation for these logarithms, you would need to use a calculator or a computational tool to evaluate the logarithmic values. Keep in mind that these values might not be exact integers.