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4. log2 4 + log4 2

Both terms in the expression are logarithms with different bases. We can use the change of base rule to express both logarithms in terms of a common base, for example, base 10. Then, we can use the sum rule of logarithms:

log2 4 + log4 2 = log10 4 / log10 2 + log10 2 / log10 4 = (log10 2^2) / log10 2 + (log10 2) / (log10 2^2) = 2 / 1 + 1 / 2 = 3

5. log 1000 + ln e^5

The first term is a logarithm in base 10, and the second term is a natural logarithm (ln, which is equivalent to log base e). We can simplify the expression using the following properties:

log 1000 = log (10^3) = 3 * log 10 ln e^5 = (log e) * 5 = 5 * 1 (since log e = 1)

Therefore, the expression becomes:

3 * log 10 + 5 * 1 = 3 log 10 + 5

6. log1/2 8

This expression represents a logarithm with base 1/2. However, logarithms with bases less than 1 are not allowed. Therefore, the expression is undefined.

7. x^2 = log 100 + 119

Take the square root of both sides:

x = ±√(log 100 + 119) ≈ ±√121 x ≈ ±11

8. log 4 + log x = log 28

Combine the logarithms on the left using the sum rule:

log (4x) = log 28 Since the bases of the logarithms are the same, we can equate the arguments:

4x = 28 x = 7

9. log (x + 3) + log x = log 28

Combine the logarithms on the left using the product rule:

log ((x + 3)x) = log 28 Since the bases of the logarithms are the same, we can equate the arguments:

(x + 3)x = 28 Expand the left side:

x^2 + 3x – 28 = 0 Factor the expression:

(x + 7)(x – 4) = 0 Therefore, x = -7 or x = 4.

Remember that the following steps should be considered when solving logarithmic equations:

1. Identify the type of logarithm (base 10, natural logarithm, etc.).
2. Consider the domain of the logarithmic function.
3. Use the properties of logarithms, such as the change of base rule, sum rule, and product rule, to manipulate the expression.
4. Convert the equation to an exponential form if necessary.
5. Solve for the variable.