Simplify the expressions in Exercises 59-62.

59.
a. [tex]\log_4 4[/tex]
b. [tex]\log_4 \sqrt{2}[/tex]
c. [tex]1.3 \log_{3} 75[/tex]
d. [tex]\log_4 16[/tex]
e. [tex]\log_3 \sqrt{3}[/tex]
f. [tex]\log_e(e)[/tex]

60.
a. [tex]2\log 3[/tex]
b. [tex]10 \log(1/2)[/tex]
c. [tex]7\log 7[/tex]
d. [tex]\log_{11} 121[/tex]
e. [tex]\log_{121} 11[/tex]
f. [tex]\log(2\sqrt{3})[/tex]

61.
a. [tex]25\log_{5} (3x)[/tex]
b. [tex]\log_{e}(e)[/tex]
c. [tex]\log_2 (\sin 2)(\sin x)[/tex]

62.
a. [tex]\log_7 7[/tex]
b. [tex]\log_2 10\sqrt{10}[/tex]
c. [tex]\log_u (2e \sin x)[/tex]

Logarithmic expressions are simplified using the properties of logarithms, such as finding the powers or applying the product rule. The given expressions are simplified step by step using these properties.

The given expressions involve logarithms, which are mathematical functions used to solve exponential equations. To simplify the expressions, you can use the properties of logarithms. Here are the steps to simplify each expression:

To simplify log4 16, we need to find the power of 4 that equals 16. Since [tex]4^2[/tex] = 16, we can write log4 16 as 2.
To simplify loge(e), we need to find the power of e that equals e. Since [tex]e^1[/tex] = e, we can write loge(e) as 1.
To simplify log2 (ein 2)(sin x), we can use the product rule of logarithms. It states that log (ab) = log a + log b. So log2 (ein 2)(sin x) = log2 (ein 2) + log2 (sin x).

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