Write In Logarithmic Form For 5, 3, And 7, 1097

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The provided instructions are a collection of various logarithmic and exponential conversion problems. To clarify and organize the task, the core assignment is to convert given equations between exponential and logarithmic forms, simplify logarithmic expressions, and solve for variables when necessary. These problems assess understanding of the fundamental relationships between exponents and logarithms, the ability to switch between forms, and simplification skills.

Paper For Above instruction

Logarithmic and exponential functions are inherently interconnected; understanding their relationships is essential for solving equations involving growth, decay, and various other phenomena in mathematics and applied sciences. The assignment tasks involve converting equations from exponential form to logarithmic form and vice versa, along with simplifying logarithmic expressions and solving for unknown variables.

Firstly, converting from exponential to logarithmic form is based on the fundamental identity: a^y = x is equivalent to log_a x = y. Using this, for the equations provided, one can transform the given exponential expressions into their corresponding logarithmic forms:

• Convert 5³ = 125: in logarithmic form, it becomes log₅ 125 = 3.
• Convert e⁷ = 1097: logarithmic form is log_e 1097 = 7. Since log_e is natural logarithm, it's often written as ln 1097 = 7.
• Convert 63 = 216: notation suggests 6³ = 216; so in logarithmic form: log₆ 216 = 3.
• Convert 4³ = 64: in logarithmic form, log₄ 64 = 3.
• Convert 125^1/3 = 5: logarithm form is log₁₂₅ 5 = 1/3.
• Convert 7³ = 343: the logarithmic form is log₇ 343 = 3.

Next, simplifying logarithmic expressions:

• Simplify log₅ 57: this expression can't be simplified further without a calculator since 57 isn't a perfect power of 5.
• Simplify log₂: typically, this refers to log₂ x. If specific value is provided, it can be simplified.
• Simplify log₂ 25: can be rewritten as log₂ (5²) = 2 log₂ 5.
• Convert log_w Q = 7: exponential form is Q = w⁷.
• Convert log₁₂₅ 1/3 = ?: is already a logarithm; if solving, the exponential form is 125^1/3 = ?.
• Simplify log₆ 216: since 6³ = 216, the logarithm simplifies to 3.
• Convert log₉ = -2: if the base is 9 and the value is missing, need additional context; but if it means log₉ x = -2, the exponential form is x = 9^-2 = 1/81.
• Simplify log₉: without a specific argument, cannot simplify further.
• Simplify log₁₀ 10: directly simplifies to 1.
• Simplify log₆: needs an argument for full simplification.
• Simplify 10 log 5: assumes common logarithm, so it equals 10 log₁₀ 5, which is 10 0.69897 ≈ 6.9897.

These exercises demonstrate core skills in manipulating and interpreting logarithmic and exponential equations. Mastery of these concepts enables solving real-world problems involving exponential growth, decay, and logarithmic scales.

References

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